You mean… TOS? :D
@ragingloli is right about the first two you mentioned. The Trouble with Tribbles is kind of a joke, but it’s worth seeing, and Mirror, Mirror is a great story, plus you get to see Spock with a goatee, and Uhura in an especially enticing costume, flirting with Sulu using a dagger. AHEM!
As for Kirk & Spock being forced to fight to the death, I’m pretty sure that was in several episodes. At least one of them was Amok Time, when Spock has to travel back to Vulcan to contend with his primal urges.
Errand of Mercy is probably my favorite episode. It’s a total Cold War drama (as were many of the episodes in the late-60s original Star Trek)—the Federation is vying for the cooperation of a planet full of pacifists. The Enterprise crew has to try to convince these pacifists that it’s worth their while to side with the Federation, rather than the predatory Klingons. I love this episode for the interactions between Kirk and the head Klingon, who has so much dignity and honor that he really should have been a Romulan.
The City on the Edge of Forever is one of the most famous episodes, and involves time travel to Depression-era America.
Plato’s Stepchildren is another great story, and featured the first interracial kiss on American television.
You definitely want to check out Space Seed, our first glimpse of Khan.
Shore Leave is fun once you get to know all the characters… They stop for shore leave on a fantasy planet where whatever you dream of just appears…
You gotta see Patterns of Force. They travel to a planet that strangely resembles Nazi Germany. Worth it just to see Spock crack on Kirk for making a convincing Nazi (and both actors are American Jews, in fact!). Also, you get to see Leonard Nimoy’s bare chest, but I don’t know if that’ll be a clincher for you. (It is for me and @Kardamom, I happen to know…)
The Corbomite Maneuver is quite famous and I’ve seen it parodied several times in popular culture, and features Ron Howard’s very strange looking young brother Clint Howard.
Finally, I’d recommend Mudd’s Women, and I, Mudd. Harry Mudd is truly larger than life, kind of a space pirate, and these are funny episodes.
Happy watching :) Now I’ve got to see the new movie myself…