OK, folks. Neozeks gets the credit for planting the idea in my head of trying to cover something we never actually see happen in Star Trek, something of far more scope and interstellar consequence than how one uses the bathroom.
Namely, how the hell new members are integrated into the Federation. In particular, how their military forces are integrated. (This first post wound up focusing on First Contact and is absurdly long. My apologies.)
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The rules of thumb posted in that linked thread apply here. []'s indicate margin notes. The perspective of the document is approximately 2383 to 2385. This document is out of character in nature.
With that, modelling assumptions:
1. The Federation has money and economics. It is, generally speaking, what those of the 21st century would call a capitalist economy, although with a good deal of governmental intervention.
2. Use the governmental structure for the UFP proposed in my Federation governmental structure thread.
3. Religion exists. Humans may be a plurality atheist (or at least of a secularist bent), but human religion exists as well. It is not a given that a species is religious or atheist.
4. Starfleet: Utilize the Starfleet Command structure and Starfleet Academy design postulated in the respective threads.
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Introduction
[This all describes a non-emergency situation (The Borg aren't coming to eat the target world, nobody's poised to invade the local space, etc, etc.) - Obviously, if circumstances are more demanding, what happens changes, or happens faster.]
So, your species has broken the warp barrier! You can travel faster than light! Congratulations!
Along with this accomplishment and all that it might do to a society, a planet, a civilization...By 2383-2385 in the Terran calendar, you're also likely faced with something else, something possibly even more worthy of a "I want this all to go away!" reaction where you climb into bed and never get out again, like a small child. Very soon after you first achieve FTL travel, people claiming not to be of your species either come out of the woodwork or contact you from ships...ships orbiting your homeworld. They claim to be not just of one alien species but many, to be the representatives of an interstellar government comprising dozens, hundreds of species even! And they'd like to say hello.
First Contact
Before one can even begin to speak of a world joining the Federation, one has to begin by thinking of First Contact. While it's possible that contact could be made by ships in space running across Federation (or other groups') ships, which is possibly the most ideal scenario, the reality is that by 2383-2385, in the Alpha Quadrant and near Beta Quadrant, it is far more likely that contact is made either by First Contact teams that had been observing on a planet, or as a result of sensor detection by Federation ships (or other sensors) of consistent FTL activity, either around a world or between worlds.
As a general rule, Federation observation teams on a planet go in before a society achieves warp travel; they do their level best to get out before FTL travel (and especially FTL sensors) are achieved, as well. Sometimes, however, events move too fast or combine with unexpected issues to make getting out beforehand impossible. In those cases, observation teams are trained and ordered to keep observing, clandestinely, to the best of their ability, and not to make First Contact if they can possibly avoid it. This is largely informed by the human experience and human intuition, though it's a point agreed with as a general principle by virtually every species (particularly within the UFP), and confirmed by inevitable occasional experience: First Contact made by in-place observation teams, particularly teams that had posed as members of the target species, tends to...seriously unnerve the target species, most particularly the leadership thereof. [This is a significant understatement. Humans, in the Trek timeline, are possibly uniquely prepared by the sheer commonality of the modern UFO mythos for the idea of extraterrestrial contact. Even so, First Contact must have seriously freaked people out when it did occur, no matter how prepared they may have thought they were for the idea. It seems unlikely that other species would react as well to the idea of extraterrestrials, and reactions would likely be even worse if the ETs emerged from within the society(/societies), as an Observation Team caught on-planet would. Add whole new levels of worse if the planet doesn't have a single government. You could try to be as non-threatening and as gentle as you please, but it would still be a horrible way to make an introduction.]
Much preferable is to extract Observation Teams without on-planet notice, posting Federation ships (at a point where on-planet sensors likely cannot detect them) to keep outsiders from barging in. (Occasionally, this requires the use of force, but it's the rare interstellar traveler who doesn't respect a Federation request/demand to keep away from a "newly arrived species". Other groups and governments, for fear of more general Federation reaction, don't usually push the matter - The UFP is really touchy about this issue.) This may seem like an unnecessary application of the Prime Directive, an interference in the development of a warp-capable species, but it has proven essential to "easing in" a society to the idea of extraterrestrial life. [They may think they're ready. You may think, for a minute, they're ready. Truth is, no species is ever truly ready to find out they aren't alone in the cosmos, and it's doubtful any ever would be.] The target world is then observed from space (by posted "picket ships" and very occasional close-in measures such as probes) until sensors capable of detecting FTL ships are developed, until the target species seems likely to step outside of their immediate area, or until other evidence arises of the target world being about to stumble upon the truth regarding extraterrestrial life. [As a rule, if the target world develops FTL sensors (which may not happen at the same time as FTL mobility, and indeed probably happens a bit later), they'll be able to see the ships watching them, confirm their existence, and react. Hence, at that point you'd best be ready to say hello, whether you like it or not. (And there are times, yes, where either the Federation, the target species, or both wish that they'd had the chance to delay that hello, whether for a short time or a long time. However, once the target species can see for itself that there are others out there, you can't avoid it, even if it's ill-timed.)]
What happens next depends greatly upon the individual target species and their circumstances, the nature of their "eligibility" for First Contact, the time between warp travel being achieved and first contact, Federation resources available, the galactic situation, the level of knowledge about the target species, whether there are still Observation Teams on-planet, and other factors. That said, there are general rules:
1. It's greatly preferred that shipborne assets make the first contact, not on-planet Observation Teams: The "aliens" being out in space is a lot less scary than the idea they've been living among you and you haven't had a clue. [OTs making the First Contact just scares the locals, and should hence be avoided if at all possible.]
2. Don't send the biggest ship you can find. Send a smaller ship. If it's briefed, trained, crewed, and equipped to specifically to say hello, even better: The Galaxy-class (for example, but replace with any larger ship) may have the most diplomatic captains and the best diplomatic facilities, but it's also incredibly huge, especially if you're relatively new to space. You may not to be able to avoid it, but try to send a small ship. It scares the locals less. If it can be prepped for the mission, say through crew training, the presence of Federation diplomatic personnel (and personnel from the Starfleet First Contact Office unit responsible for the target species 'portfolio'), command crew briefing by the First Contact Office (who should also attempt to brief the rest of the ship's crew), etc., all the better. If you can't mobilize Diplomatic Service or relevant FCO personnel to physically be there, have them "on speed dial" as well as Starfleet Command. This can be essential.
3. If there are still OTs on-planet that haven't been extracted, they aren't the ones making First Contact, and you can do so without the locals noticing, try to get them out before you send the first signal saying hello. This may not always be possible, but it's worth delaying the first contact if you can stealthily extract the OTs, debrief them, and reverse cosmetic alterations before the target species is contacted. They may have information not otherwise available, as well as skills (such as target language ability without UT usage - UTs are better than nothing, but for various reasons, newly contacted species tend to like people who speak their own language "naturally") nobody else has. If you cannot delay first contact, OTs are trained to stay clandestine and continue observing until they can be extracted if at all possible, not revealing their presence unless they must. [Or until they're discovered as extraterrestrials by the locals.]
4. First Contact should occur only with a limited spectrum of species persons. Leadership elements are always preferred, followed by military personnel (who tend to be listened to by the leadership), followed by security forces, followed only then by scientists. Avoid it with some random person. [The Vulcans picking Cochrane and friends to say hello to first was perhaps the worst possible example.] In the event there isn't a single world government, contact the confirmed "Great Powers" first. If they have an organization such as the United Nations, this is sometimes best to use as a conduit, if the planetary political situation warrants. Otherwise, you have two options: Simultaneous contact of the Great Powers, or far less preferably, picking one nation (preferably a Great Power) to contact, ease into comfort with the idea of extraterrestrial life, and use to "spread the news", with the utmost discretion, to the other nations. (If there is a "superpower", or multiple superpowers, either talk to them first or talk to them at the same conversation as with the Great Powers.) Never, ever make contact to or through media if you can avoid it. For that matter, try not to even merit the notice of the media before contacting the target world's leadership. It may be the case that the target society's leadership (particularly if there isn't a single world government) decides not to reveal the existence of ETs to the general public just yet, for whatever reasons - this may be a very good idea. Doubters are asked to imagine the likely reaction had extraterrestrials been discovered and confirmed during, for example, most of Earth's 20th Century in sociological terms, or actually during World War III (instead of after World War III). If there are religious hierarchies or even just religions in the target society/societies - swallow your own views on the topic of religious belief, whatever they may be, and accept the fact of the target society's beliefs. (Now is absolutely not the time to tell the target species that there is, or is not, a God. Period! Do not even discuss the topic if you can avoid it, certainly do not bring it up, and if they bring it up, duck the question. They are, not to put too fine a point on it, going to go through enough as a species.) Contact them along with political powers. (To use a 21st Century human example, if you're making First Contact with the UN, or even just the superpowers outside of the UN, take the time to get representatives of the Pope and other religious hierarchs and hierarchies included on the conversation. (You can't expect to "get everybody in the room" on even a moderately religiously-diverse planet, but where there are hierarchical structures and "notables", invite them along with political/military powers.) It can sometimes be enormously helpful in getting the news of "life out there" to "go down easy" when the time comes if the major religions are also informed, actively involved in "preparing the ground", and generally not shut out of the conversation.) If you can, get leaders (or at least representatives such as diplomats of ambassadorial rank) in the same physical room at the same time - if that isn't possible, use secure communications systems (preferably with video capabilities) to get everyone in the same conversation.
5. Only with the active concurrence of the target world's leadership should the existence of extraterrestrial life be revealed to the general population. Period. Mass societal hysteria can easily result if you "go over the heads" of the leadership. If the leadership is not ready, for whatever reasons, then offer to limit contact to the governments of those powers already aware (or who become aware) of the existence of extraterrestrial life, with the explicit goal of making them comfortable with extraterrestrial contact and at some point revealing the fact of extraterrestrial life to the general population. Respect the political evolution of the planetary structure: Do not contact governments not generally recognized (a good rule is that if none of the great powers will talk to them, you shouldn't either unless circumstances are obviously in flux). After the initial conversation with global leadership, back off and let them decide what to do, careful to avoid even the appearance or implication of extraterrestrial pressure or "dictates". Do advise them that Federation starships and sensors will, to the best of their ability, continue to "fence them off" from interstellar visitors until they decide to step into the galactic community (and do make clear Federation preference that such a "debut" wait until the target world's societies are generally aware of extraterrestrial life; gently but firmly discourage "forward runners" from "getting ahead" of others, and explain openly to leaders that this is to encourage the planetary leadership strata to adopt a single policy, whatever that policy may be), as a precautionary measure.
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There's more but the buffer won't take the whole thing at once. Please wait for the whole thing.

Namely, how the hell new members are integrated into the Federation. In particular, how their military forces are integrated. (This first post wound up focusing on First Contact and is absurdly long. My apologies.)
---
The rules of thumb posted in that linked thread apply here. []'s indicate margin notes. The perspective of the document is approximately 2383 to 2385. This document is out of character in nature.
With that, modelling assumptions:
1. The Federation has money and economics. It is, generally speaking, what those of the 21st century would call a capitalist economy, although with a good deal of governmental intervention.
2. Use the governmental structure for the UFP proposed in my Federation governmental structure thread.
3. Religion exists. Humans may be a plurality atheist (or at least of a secularist bent), but human religion exists as well. It is not a given that a species is religious or atheist.
4. Starfleet: Utilize the Starfleet Command structure and Starfleet Academy design postulated in the respective threads.
---
Introduction
[This all describes a non-emergency situation (The Borg aren't coming to eat the target world, nobody's poised to invade the local space, etc, etc.) - Obviously, if circumstances are more demanding, what happens changes, or happens faster.]
So, your species has broken the warp barrier! You can travel faster than light! Congratulations!
Along with this accomplishment and all that it might do to a society, a planet, a civilization...By 2383-2385 in the Terran calendar, you're also likely faced with something else, something possibly even more worthy of a "I want this all to go away!" reaction where you climb into bed and never get out again, like a small child. Very soon after you first achieve FTL travel, people claiming not to be of your species either come out of the woodwork or contact you from ships...ships orbiting your homeworld. They claim to be not just of one alien species but many, to be the representatives of an interstellar government comprising dozens, hundreds of species even! And they'd like to say hello.
First Contact
Before one can even begin to speak of a world joining the Federation, one has to begin by thinking of First Contact. While it's possible that contact could be made by ships in space running across Federation (or other groups') ships, which is possibly the most ideal scenario, the reality is that by 2383-2385, in the Alpha Quadrant and near Beta Quadrant, it is far more likely that contact is made either by First Contact teams that had been observing on a planet, or as a result of sensor detection by Federation ships (or other sensors) of consistent FTL activity, either around a world or between worlds.
As a general rule, Federation observation teams on a planet go in before a society achieves warp travel; they do their level best to get out before FTL travel (and especially FTL sensors) are achieved, as well. Sometimes, however, events move too fast or combine with unexpected issues to make getting out beforehand impossible. In those cases, observation teams are trained and ordered to keep observing, clandestinely, to the best of their ability, and not to make First Contact if they can possibly avoid it. This is largely informed by the human experience and human intuition, though it's a point agreed with as a general principle by virtually every species (particularly within the UFP), and confirmed by inevitable occasional experience: First Contact made by in-place observation teams, particularly teams that had posed as members of the target species, tends to...seriously unnerve the target species, most particularly the leadership thereof. [This is a significant understatement. Humans, in the Trek timeline, are possibly uniquely prepared by the sheer commonality of the modern UFO mythos for the idea of extraterrestrial contact. Even so, First Contact must have seriously freaked people out when it did occur, no matter how prepared they may have thought they were for the idea. It seems unlikely that other species would react as well to the idea of extraterrestrials, and reactions would likely be even worse if the ETs emerged from within the society(/societies), as an Observation Team caught on-planet would. Add whole new levels of worse if the planet doesn't have a single government. You could try to be as non-threatening and as gentle as you please, but it would still be a horrible way to make an introduction.]
Much preferable is to extract Observation Teams without on-planet notice, posting Federation ships (at a point where on-planet sensors likely cannot detect them) to keep outsiders from barging in. (Occasionally, this requires the use of force, but it's the rare interstellar traveler who doesn't respect a Federation request/demand to keep away from a "newly arrived species". Other groups and governments, for fear of more general Federation reaction, don't usually push the matter - The UFP is really touchy about this issue.) This may seem like an unnecessary application of the Prime Directive, an interference in the development of a warp-capable species, but it has proven essential to "easing in" a society to the idea of extraterrestrial life. [They may think they're ready. You may think, for a minute, they're ready. Truth is, no species is ever truly ready to find out they aren't alone in the cosmos, and it's doubtful any ever would be.] The target world is then observed from space (by posted "picket ships" and very occasional close-in measures such as probes) until sensors capable of detecting FTL ships are developed, until the target species seems likely to step outside of their immediate area, or until other evidence arises of the target world being about to stumble upon the truth regarding extraterrestrial life. [As a rule, if the target world develops FTL sensors (which may not happen at the same time as FTL mobility, and indeed probably happens a bit later), they'll be able to see the ships watching them, confirm their existence, and react. Hence, at that point you'd best be ready to say hello, whether you like it or not. (And there are times, yes, where either the Federation, the target species, or both wish that they'd had the chance to delay that hello, whether for a short time or a long time. However, once the target species can see for itself that there are others out there, you can't avoid it, even if it's ill-timed.)]
What happens next depends greatly upon the individual target species and their circumstances, the nature of their "eligibility" for First Contact, the time between warp travel being achieved and first contact, Federation resources available, the galactic situation, the level of knowledge about the target species, whether there are still Observation Teams on-planet, and other factors. That said, there are general rules:
1. It's greatly preferred that shipborne assets make the first contact, not on-planet Observation Teams: The "aliens" being out in space is a lot less scary than the idea they've been living among you and you haven't had a clue. [OTs making the First Contact just scares the locals, and should hence be avoided if at all possible.]
2. Don't send the biggest ship you can find. Send a smaller ship. If it's briefed, trained, crewed, and equipped to specifically to say hello, even better: The Galaxy-class (for example, but replace with any larger ship) may have the most diplomatic captains and the best diplomatic facilities, but it's also incredibly huge, especially if you're relatively new to space. You may not to be able to avoid it, but try to send a small ship. It scares the locals less. If it can be prepped for the mission, say through crew training, the presence of Federation diplomatic personnel (and personnel from the Starfleet First Contact Office unit responsible for the target species 'portfolio'), command crew briefing by the First Contact Office (who should also attempt to brief the rest of the ship's crew), etc., all the better. If you can't mobilize Diplomatic Service or relevant FCO personnel to physically be there, have them "on speed dial" as well as Starfleet Command. This can be essential.
3. If there are still OTs on-planet that haven't been extracted, they aren't the ones making First Contact, and you can do so without the locals noticing, try to get them out before you send the first signal saying hello. This may not always be possible, but it's worth delaying the first contact if you can stealthily extract the OTs, debrief them, and reverse cosmetic alterations before the target species is contacted. They may have information not otherwise available, as well as skills (such as target language ability without UT usage - UTs are better than nothing, but for various reasons, newly contacted species tend to like people who speak their own language "naturally") nobody else has. If you cannot delay first contact, OTs are trained to stay clandestine and continue observing until they can be extracted if at all possible, not revealing their presence unless they must. [Or until they're discovered as extraterrestrials by the locals.]
4. First Contact should occur only with a limited spectrum of species persons. Leadership elements are always preferred, followed by military personnel (who tend to be listened to by the leadership), followed by security forces, followed only then by scientists. Avoid it with some random person. [The Vulcans picking Cochrane and friends to say hello to first was perhaps the worst possible example.] In the event there isn't a single world government, contact the confirmed "Great Powers" first. If they have an organization such as the United Nations, this is sometimes best to use as a conduit, if the planetary political situation warrants. Otherwise, you have two options: Simultaneous contact of the Great Powers, or far less preferably, picking one nation (preferably a Great Power) to contact, ease into comfort with the idea of extraterrestrial life, and use to "spread the news", with the utmost discretion, to the other nations. (If there is a "superpower", or multiple superpowers, either talk to them first or talk to them at the same conversation as with the Great Powers.) Never, ever make contact to or through media if you can avoid it. For that matter, try not to even merit the notice of the media before contacting the target world's leadership. It may be the case that the target society's leadership (particularly if there isn't a single world government) decides not to reveal the existence of ETs to the general public just yet, for whatever reasons - this may be a very good idea. Doubters are asked to imagine the likely reaction had extraterrestrials been discovered and confirmed during, for example, most of Earth's 20th Century in sociological terms, or actually during World War III (instead of after World War III). If there are religious hierarchies or even just religions in the target society/societies - swallow your own views on the topic of religious belief, whatever they may be, and accept the fact of the target society's beliefs. (Now is absolutely not the time to tell the target species that there is, or is not, a God. Period! Do not even discuss the topic if you can avoid it, certainly do not bring it up, and if they bring it up, duck the question. They are, not to put too fine a point on it, going to go through enough as a species.) Contact them along with political powers. (To use a 21st Century human example, if you're making First Contact with the UN, or even just the superpowers outside of the UN, take the time to get representatives of the Pope and other religious hierarchs and hierarchies included on the conversation. (You can't expect to "get everybody in the room" on even a moderately religiously-diverse planet, but where there are hierarchical structures and "notables", invite them along with political/military powers.) It can sometimes be enormously helpful in getting the news of "life out there" to "go down easy" when the time comes if the major religions are also informed, actively involved in "preparing the ground", and generally not shut out of the conversation.) If you can, get leaders (or at least representatives such as diplomats of ambassadorial rank) in the same physical room at the same time - if that isn't possible, use secure communications systems (preferably with video capabilities) to get everyone in the same conversation.
5. Only with the active concurrence of the target world's leadership should the existence of extraterrestrial life be revealed to the general population. Period. Mass societal hysteria can easily result if you "go over the heads" of the leadership. If the leadership is not ready, for whatever reasons, then offer to limit contact to the governments of those powers already aware (or who become aware) of the existence of extraterrestrial life, with the explicit goal of making them comfortable with extraterrestrial contact and at some point revealing the fact of extraterrestrial life to the general population. Respect the political evolution of the planetary structure: Do not contact governments not generally recognized (a good rule is that if none of the great powers will talk to them, you shouldn't either unless circumstances are obviously in flux). After the initial conversation with global leadership, back off and let them decide what to do, careful to avoid even the appearance or implication of extraterrestrial pressure or "dictates". Do advise them that Federation starships and sensors will, to the best of their ability, continue to "fence them off" from interstellar visitors until they decide to step into the galactic community (and do make clear Federation preference that such a "debut" wait until the target world's societies are generally aware of extraterrestrial life; gently but firmly discourage "forward runners" from "getting ahead" of others, and explain openly to leaders that this is to encourage the planetary leadership strata to adopt a single policy, whatever that policy may be), as a precautionary measure.
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There's more but the buffer won't take the whole thing at once. Please wait for the whole thing.