Don't stone me for being a heretic, but I thought since there was a thread like this for all the other shows, we might as well join in the fun. I know there's not a whole lot you can do differently to improve the sacred cow, but imagine for a moment you're Gene Roddenberry. You're figuring out what this 'Wagon Train to the Stars' is going to be like. What would your crew have been like? What would your ship have been like? I encourage you to use dropped plot elements or characters as well as come up with your own ideas. And remember, it's the 60s. 
My take, set in 2165, approximately 200 years in the 'future.'
The ship:
U.S.S. Yorktown -
Starship-class (a term referring to its size and range, not a particular design) vessel charged with patrol and exploration on the outer realms of Federation space. The Federation is a looser alliance of sovereign member worlds, more like the United Nations. The Yorktown is an Earth ship, and as such operates under the joint authority of the United Earth Space Probe Agency (a combined military-exploration service that has its roots in a multi-national NASA) and the Federation Starfleet (a NATO-type force comprised of rotating forces from various members' militaries.) Technically, the Yorktown crew are UESPA officers, not Starfleet officers. Some UESPA missions are allowed to override Starfleet missions, and vice versa.
The Yorktown is 474 feet long and carries a crew of 203. It is comprised of a saucer section, cigar-shaped secondary hull, and twin outboard 'warp power units' which not only generate the ship's main power via a small matter/antimatter reactor in each, but also provide the ship's ability to warp space, allowing faster than light travel without time dilation effects. Power from the twin engines is routed via their support pylons into the secondary hull, where converters channel this into electrical energy for the ship (meaning no warp core in the hull.) Sublight power is provided by 'impulse engines,' Newtonian rockets which operate from fusion-derived power. These fusion reactors may also power the ship in an emergency for a limited time. Impulse engines may also provide near-lighstpeed velocity in an emergency, with a danger of time dilation. Being assigned to isolated long range missions, the Yorktown is supplied to provide for basic needs of the crew and efficiently recycle its waste, but features little more than basic rest facilities. As such, shore leave on inhabitable planets is key to maintaining crew rest and morale. The ship's weapons consist of 'ace (amplified coherent energy) guns' which are also carried as sidearms, as well as 'photon torpedoes' which are largely unavoidable and carry a small combination of matter and antimatter for devastating kill-shots.
Main Crew:
Captain Robert T. April (Jack Lord)
April is a veteran of UESPA service and a maverick captain, having served with distinction on various ships. This is not his first command, having first served as commander of two smaller ships, the first a small corvette when he was a Lieutenant Commander. April is brave, headstrong, and a leader by deed as well as word. His very persona exudes confidence. April has a wife and children back home, and he can occasionally be seen recording messages for them, or listening to their own messages.
1st Officer/Helmsman Cmmdr. Elizabeth Morgan (Majel Barrett)
The ship's first officer and helmsman, as well as the most experienced officer aboard next to April, she is April's 'Number One.' Morgan is native to the Earth colony Cygnet XIV, founded prior to development of warp drive, whose culture developed in the interim as a matriarchy. As such, Morgan is a fair representative of her civilization but seems somehow out of place aboard the ship's equally headstrong men. Morgan is normally left in command when the Captain leads landing parties, and is always responsible for the security of the ship's mission and the safety of her crew. Morgan is secretly interested in Engineer Pike, a fact she would surely never reveal. Indeed, as member of one of her world's most prominent families, she is expected to return to Cygnet at some point to complete an arranged marriage.
2nd Officer/Engineer Lt. Cmmdr. Christopher Pike (Jeffrey Hunter)
A thoughtful, somewhat introverted officer who has his eye on getting a command of his own but is content for now to be April's engineer. Pike grew up in Mojave (a former desert) on Earth, always longing to see the unknown. He is relentlessly curious, and something of a bookworm, but nonetheless brave and dedicated. He is probably the ship's ladies' man.
Chief Surgeon Lt. Cmmdr Philip Boyce (DeForest Kelley)
A cantankerous former practicing doctor and friend to the captain. He leaves a wife and adult daughter behind, and sought the space service after feeling that Earth-bound medicine lacked any challenge for him. He distrusts anything 'new' and believes fundamentally in the power of the human will. He has a bantering but respectful relationship with Mister Spock, and is close friends with all of his superior officers.
Science Officer Lt. Spock (Mark Lenard)
A native of the planet Vulcania, Spock is one of few aliens to join the Earth UESPA, owing perhaps to being 1/8th human himself. His devilish appearance and slightly reddish complexion belies his relatively calm demeanor. Though relatively young, Spock is perhaps the most knowlegeable man aboard. Most people from Vulcania are known for their logical minds, however logical does not mean emotionless. A young man is still a young man, and Spock is known for his occasional excited outbursts.
Rotating Crew (Day Players):
Communications Chief Lt. Montgomery Scott (James Doohan)
A Scottsman as the name might imply, 'Scotty' is an elder master of various technologies and a product of the engineering programs of UESPA, but has proven particularly adept as a linguistics and as a communications specialist. He has shown a proven record of deciphering enemy transmissions and inventing clever new uses of the ship's radios to break through barriers, both physical and cultural. As a frequent visitor to the engine rooms, he is a close friend of Pike, and a drinking pal of Boyce.
Navigator Lt. Isoroku Sulu (George Takei)
An astrophysicist who wears a blue division tunic, Sulu usually sits opposite Number One at the helm console, tracking the ship's course and speed relative to known space. Inherently curious, Sulu is a man of many talents and hobbies, and also extremely knowlegeable when it comes to spatial phenomena. Sulu is also responsible for relaying orders (via his control panel) for the firing of the ship's weapons.
Yeoman Upenda Uhura (Nichelle Nichols)
The Captain's yeoman, following the naval tradition. Captain April was initially uncomfortable with having a female yeoman for modesty reasons, given their close interaction but has since come to respect and admire the young officer. Uhura is gifted in song, and often serenades the crew in her off time, sometimes accompanied by the musical talents of Chekhov.
Security Chief Jose Tyler (John Saxon)
The ship's resident Latin hothead, Tyler inherited his Brazilian mother's legendary temper and is just as quick to act when necessary. He (or occasionally one of his men) usually accompanies the Captain on landing teams to ensure his safety.
Relief Science Officer Piotr Chekhov (Peter Duel)
When Spock is away from his station, young Russian genius Chekhov is responsible for the manning of the ship's sensor scopes. Chekhov is a proud Russian, who often references being a descendant of the famed author Anton Chekhov. He is talented at playing several different musical instruments, including the Vulcanian lyre, a subject which has allowed a bond to form between himself and his superior science officer.
Chief Nurse Janice Rand (Sally Kellerman)
Boyce's head nurse, who holds a possibly requited attraction to Mister Spock.

My take, set in 2165, approximately 200 years in the 'future.'
The ship:
U.S.S. Yorktown -
Starship-class (a term referring to its size and range, not a particular design) vessel charged with patrol and exploration on the outer realms of Federation space. The Federation is a looser alliance of sovereign member worlds, more like the United Nations. The Yorktown is an Earth ship, and as such operates under the joint authority of the United Earth Space Probe Agency (a combined military-exploration service that has its roots in a multi-national NASA) and the Federation Starfleet (a NATO-type force comprised of rotating forces from various members' militaries.) Technically, the Yorktown crew are UESPA officers, not Starfleet officers. Some UESPA missions are allowed to override Starfleet missions, and vice versa.
The Yorktown is 474 feet long and carries a crew of 203. It is comprised of a saucer section, cigar-shaped secondary hull, and twin outboard 'warp power units' which not only generate the ship's main power via a small matter/antimatter reactor in each, but also provide the ship's ability to warp space, allowing faster than light travel without time dilation effects. Power from the twin engines is routed via their support pylons into the secondary hull, where converters channel this into electrical energy for the ship (meaning no warp core in the hull.) Sublight power is provided by 'impulse engines,' Newtonian rockets which operate from fusion-derived power. These fusion reactors may also power the ship in an emergency for a limited time. Impulse engines may also provide near-lighstpeed velocity in an emergency, with a danger of time dilation. Being assigned to isolated long range missions, the Yorktown is supplied to provide for basic needs of the crew and efficiently recycle its waste, but features little more than basic rest facilities. As such, shore leave on inhabitable planets is key to maintaining crew rest and morale. The ship's weapons consist of 'ace (amplified coherent energy) guns' which are also carried as sidearms, as well as 'photon torpedoes' which are largely unavoidable and carry a small combination of matter and antimatter for devastating kill-shots.
Main Crew:
Captain Robert T. April (Jack Lord)
April is a veteran of UESPA service and a maverick captain, having served with distinction on various ships. This is not his first command, having first served as commander of two smaller ships, the first a small corvette when he was a Lieutenant Commander. April is brave, headstrong, and a leader by deed as well as word. His very persona exudes confidence. April has a wife and children back home, and he can occasionally be seen recording messages for them, or listening to their own messages.
1st Officer/Helmsman Cmmdr. Elizabeth Morgan (Majel Barrett)
The ship's first officer and helmsman, as well as the most experienced officer aboard next to April, she is April's 'Number One.' Morgan is native to the Earth colony Cygnet XIV, founded prior to development of warp drive, whose culture developed in the interim as a matriarchy. As such, Morgan is a fair representative of her civilization but seems somehow out of place aboard the ship's equally headstrong men. Morgan is normally left in command when the Captain leads landing parties, and is always responsible for the security of the ship's mission and the safety of her crew. Morgan is secretly interested in Engineer Pike, a fact she would surely never reveal. Indeed, as member of one of her world's most prominent families, she is expected to return to Cygnet at some point to complete an arranged marriage.
2nd Officer/Engineer Lt. Cmmdr. Christopher Pike (Jeffrey Hunter)
A thoughtful, somewhat introverted officer who has his eye on getting a command of his own but is content for now to be April's engineer. Pike grew up in Mojave (a former desert) on Earth, always longing to see the unknown. He is relentlessly curious, and something of a bookworm, but nonetheless brave and dedicated. He is probably the ship's ladies' man.
Chief Surgeon Lt. Cmmdr Philip Boyce (DeForest Kelley)
A cantankerous former practicing doctor and friend to the captain. He leaves a wife and adult daughter behind, and sought the space service after feeling that Earth-bound medicine lacked any challenge for him. He distrusts anything 'new' and believes fundamentally in the power of the human will. He has a bantering but respectful relationship with Mister Spock, and is close friends with all of his superior officers.
Science Officer Lt. Spock (Mark Lenard)
A native of the planet Vulcania, Spock is one of few aliens to join the Earth UESPA, owing perhaps to being 1/8th human himself. His devilish appearance and slightly reddish complexion belies his relatively calm demeanor. Though relatively young, Spock is perhaps the most knowlegeable man aboard. Most people from Vulcania are known for their logical minds, however logical does not mean emotionless. A young man is still a young man, and Spock is known for his occasional excited outbursts.
Rotating Crew (Day Players):
Communications Chief Lt. Montgomery Scott (James Doohan)
A Scottsman as the name might imply, 'Scotty' is an elder master of various technologies and a product of the engineering programs of UESPA, but has proven particularly adept as a linguistics and as a communications specialist. He has shown a proven record of deciphering enemy transmissions and inventing clever new uses of the ship's radios to break through barriers, both physical and cultural. As a frequent visitor to the engine rooms, he is a close friend of Pike, and a drinking pal of Boyce.
Navigator Lt. Isoroku Sulu (George Takei)
An astrophysicist who wears a blue division tunic, Sulu usually sits opposite Number One at the helm console, tracking the ship's course and speed relative to known space. Inherently curious, Sulu is a man of many talents and hobbies, and also extremely knowlegeable when it comes to spatial phenomena. Sulu is also responsible for relaying orders (via his control panel) for the firing of the ship's weapons.
Yeoman Upenda Uhura (Nichelle Nichols)
The Captain's yeoman, following the naval tradition. Captain April was initially uncomfortable with having a female yeoman for modesty reasons, given their close interaction but has since come to respect and admire the young officer. Uhura is gifted in song, and often serenades the crew in her off time, sometimes accompanied by the musical talents of Chekhov.
Security Chief Jose Tyler (John Saxon)
The ship's resident Latin hothead, Tyler inherited his Brazilian mother's legendary temper and is just as quick to act when necessary. He (or occasionally one of his men) usually accompanies the Captain on landing teams to ensure his safety.
Relief Science Officer Piotr Chekhov (Peter Duel)
When Spock is away from his station, young Russian genius Chekhov is responsible for the manning of the ship's sensor scopes. Chekhov is a proud Russian, who often references being a descendant of the famed author Anton Chekhov. He is talented at playing several different musical instruments, including the Vulcanian lyre, a subject which has allowed a bond to form between himself and his superior science officer.
Chief Nurse Janice Rand (Sally Kellerman)
Boyce's head nurse, who holds a possibly requited attraction to Mister Spock.