As far as favorites:
original series: several in fact, but two that stand out are "The Menagerie" and "The Doomsday Machine". Also, so the 3rd season is not left out I'd pick "The Tholian Web"
animated series: "The Slaver Weapon" as it depicts my species



---ok, in all seriousness it is a great episode
TNG: "Q, Who"--the Borg were one of my favorite villains and at this early stage they seemed invincible, and mysterious.
DS9: "The Search"--this episode was very unusual, almost had a theatrical feel to it. Also it was one of those episodes I was waiting to see for years, I had seen later episodes with the Dominion and because I missed this episode when it aired I was very curious about how that all started. It felt so sweet when I finally got to see it on video and it didn't disappoint.
Voyager: "Dark Frontier"--one of the few episodes I saw as a first run. I remember thinking it was going to run over it's hour until I realized it was a feature length episode. But I loved esp. the flashback scenes. I also have to throw out there "The Thaw" (the episode with the 'clown'). As a horror movie fan I really liked that episode and it's delving into 'fear.'
Enterprise: "Impulse"--I'm a horror film fan--need I say more
Movies: The Motion Picture--always has been my favorite, always will
As far as Discovery, I'm still in my first run. But up to this point halfway through the 1st season I'd tentatively pick "Magic to Make the Sanest Man Go Mad" partly because I always liked those weird time change/loop episodes. But that may change as time goes on.