This is what I have for the Original Series so far. Some of the episodes introduce characters or plots or themes while others are just great episodes. One of the failings of TOS, from what I remember, is a lack of major character development. If I've missed episodes that are stand outs in advancing characters or whatnot, give your own input. I haven't seen these episodes for quite some time and have been relying on the infallible internet to refresh my memory.
1.00 The Cage
1.01 Where No Man Has Gone Before
1.02 The Corbomite Maneuver
These three episodes introduce us to all of the characters and give a great look at what Star Trek is all about.
1.03 Mudd's Women - This is the first of the Harry Mudd episodes. I don't recall liking the character and the episodes much.
1.06 The Naked Time - This episode is kind of weak in my opinion but it does advance the relationship between Kirk and Rand and introduces Nurse Chapel. The relationship doesn't really go anywhere so the Kirk/Rand episodes can probably be skipped.
1.08 Balance of Terror - This is our first look at Romulans and is just a great episode.
1.11 Miri - This episode advances the Kirk/Rand relationship but seems otherwise kind of forgettable.
1.12 The Conscience of the King - Rand leaves the show in this episode leaving the developing relationship between her and Kirk unresolved.
1.13 The Galileo Seven
1.15-16 The Menagerie, Parts I&II - These episodes return to Captain Pike and the events begun in The Cage.
1.24 Space Seed - KHAAAAAAAAN!!! Oh wait, that comes later.
1.26 The Devil in the Dark
1.27 Errand of Mercy - The Klingons show up for the first time here.
1.28 The City on the Edge of Forever
2.01 Catspaw - This episode adds Chekov to the crew.
2.02 Metamorphosis - Zefrham Cochrane is in this one and we learn some history unless I do not remember correctly.
2.03 Friday's Child - This one has Klingons in it but I don't know how much it moves forward the Kilingon/Federation story.
2.05 Amok Time - Not only is this a great episode it shows a lot about Vulcan culture and advances the friendship between Kirk and Spock and advances the relationship between Spock and Chapel.
2.06 The Doomsday Machine
2.10 Mirror, Mirror - This is the first time we see the Mirror Universe and it comes back a bunch in Deep Space Nine from what I hear. The Deadly years
2.11The Deadly Years -The Romulans are back.
2.12 I, Mudd - Harry Mudd is back.
2.13 The Trouble with Tribbles - I have a pet tribble that hangs out on my computer desk.
2.16 A Private Little War - More Klingons but like Friday's Child I don't know how necessary it is.
3.02 Elaan of Troyius - See above.
3.04 The Enterprise Incident - Not only is this a great episode but it also has Romulans.
3.09 The Tholian Web - According to Memory Alpha this is a Mirror Universe episode by way of Enterprise's In a Mirror, Darkly.
3.11 Day of the Dove - More Klingons.
This list makes the Original Series a mere 28 episodes. Some of them seem skippable if the Harry Mudd storyline or the Kirk/Rand story don't strike your fancy.
I know this list probably needs a lot of work. What do you think should be added or removed?