Let’s do these two together, since they’re so well connected.
The only episode of The Animated Series on this list, Yesteryear is a classic and a great hook for introducing someone to that show. The story, so human and tender, served as the inspiration for the Spock backstory in the wildly successful 2009 reboot. I particularly love the fat little Vulcan child in this.
For a show that’s supposed to be about a space navy and not a space diplomatic corps, Trek does an awful lot with space diplomats. And Journey to Babel is the episode that started it all. The introduction of Spock’s parents (ahem) humanize the character and prove that he’s more interesting than anyone else on the show. (We didn’t meet Kirk’s parents until 2009, and they play no real role in the movie besides dying and giving birth.)
JtB also introduces "Live long and prosper," the second-most famous line from all of Star Trek.