Let's do the 3 episodes thing - if I like it enough I'll move on to the rest. Thanks man!
I'm going to make some recommendations too, I tried to find a mix of episodes, some are dialog heavy bottle shows, some have more action, some have more sci-fi premises (other than being set in space) and some go into an almost comedic direction.
TOS:
Season 1, episode 14: Balance of Terror
Season 1, episode 28: The City on the Edge of Forever
Season 2, episode 04: Mirror, Mirror
TNG:
Season 2, episode 09: The Measure of a Man
Season 3, episode 15: Yesterday's Enterprise
Season 6, episodes 10/11: Chain of Command
DS9:
Season 1, episode 19: Duet
Season 2, episode 14: Whispers
Season 5, episode 06: Trials and Tribble-ations
VOY:
Season 2, episode 21: Deadlock
Season 5, episode 12: Bride of Chaotica
Season 5, episode 22: Someone to Watch Over Me
ENT:
Season 3, episode 10: Similitude
Season 4, episode 17: Bound
Season 4, episodes 18/19: In a Mirror, Darkly