A few suggestions, if what you're looking for is sort of how things got started.
The second pilot, Where No Man Has Gone Before. Kirk is the captain, Spock is there, but the rest of the crew isn't there yet, and it's interesting to see how the show changed from WNMHGB.
The Menagerie, a two part episode which includes footage from the first pilot. It gives Star Trek the feel of an actual world with a history and Spock a backstory.
Balance of Terror - where the the Romulans appear.
Errand of Mercy - where the Klingons appear.
Amok Time - which has Vulcan history and, um, interesting mating rituals.
I'd also add "Day of the Dove" for more Klingons and the delicious "Enterprise Incident" for more Romulans and "Journey to Babel" for Spock's background and Vulcans in general.
However if you really want to gain an understanding of TOS I think you'd just be better off going through Season 1 in order. Cherry-picking excellent episodes won't give you the nuances of the characters and their relationships. You need to see how they normally interact.
If you don't like Season 1 of TOS then I don't think you're going to like TOS at all.