Okay, for TOS, the best eps include:
"Errand of Mercy" (first appearance of the Klingons)
"Balance of Terror" (enter the Romulans)
"Journey to Babel"
"Amok Time"
"Devil in the Dark"
"Mirror, Mirror" (first appearance of Mirror Universe)
"City on the Edge of Forever"
"The Trouble with Tribbles"
"The Doomsday Machine"
"Arena"
I'll let somebody else tackle the latter-day series.