TNG, season 1:
"The Big Goodbye":
This minithon started rather auspiciously with the greatest hour of television ever put to film. I'm not exaggerating. It's basically my favorite thing ever. (I just learned from memory alpha that it won an Emmy! For costume design. Damn right.)
Here they all are in their Emmy-winning costumes, before things start to go wrong, being the biggest dorks imaginable.
This is the first holodeck episode, so Picard and Data and Beverly are really excited about it. Not surprisingly, it's also the first catastrophic holodeck malfunction episode, where they can't get out of the holodeck and the fail-safes are off and somebody gets shot. It's high-risk entertainment. At the end of the episode, Dixon Hill's awesome cop friend starts pondering the nature of his existence after being convinced that his world is an illusion ("What happens to this world when you leave? Are my wife and kids still waiting for me at home?"). As a kid, I always imagined this as a meta commentary about the fictional worlds created on TV shows, and decided this meant that yes, Awesome Cop's family was waiting for him at home while the holodeck was off, and yes, Picard and the rest of the Enterprise crew really existed and were having adventures even when not actively on my TV.
Baby Face Riker asks the holographic Minuet 'how far' their relationship can go, if you know what I mean, and I think you do.
Continuity! The Enterprise is at Starbase Something-or-Other to undergo repairs, including on the holodeck, because it got all death-trap-y a few episodes earlier in "The Big Goodbye." Because of the influence of the Bynars who are fixing/hijacking the ship, the holodeck generates Riker's perfect woman, a sultry jazz-loving pickup-line-appreciating brunette who also happens to be a self-aware hologram. Picard is endlessly impressed over her technology and how she interfaces with the ship's library to adapt herself to new situations. Riker is totally smitten with her and is a little heartbroken when her program is lost by the end of the episode. Minuet is awesome, and I love that later in the series she's mentioned again as Riker's one-that-got-away.
Best non-holodeck moments of this episode include Tasha and Worf going off to play Parisses Squares, just because I love Tasha. And Riker being a classic toolbag in his "a blind man teaching an android how to paint! LOL." moment. Uggghhhh. Riker, grow a beard.
(And because every episode apparently had some big impact on my childhood development, this one made me decide I loved jazz, and then I borrowed a bunch of music from the library to study it. FOR RIKER.)
TNG, season 2:
"Elementary, Dear Data":
This episode is about Data, but I couldn't resist a screencap of these outfits.
This is one of my sister's favorite episodes. It's a lot of fun! This time, they one-up the previous holodeck malfunction storyline - people can leave the holodeck, but the holodeck program controls the rest of the ship. I can only assume that after each of these malfunctions someone redesigns the holodecks to stop the exact same problem from happening again. In this case, Geordi basically gave a holodeck character sentience by accident, which seems like it shouldn't be encouraged, especially when it's legendary Sherlock Holmes nemesis Professor Moriarty (who is AWESOME in this). TNG, always obsessed with The Big Questions, continues its inquiry into whether holodeck characters are in any way "real." Moriarty realizing that he both is and isn't is amazing to watch.
To deviate a bit from holodeck glee: A lot of people point to this episode as a reason to hate Pulaski, because she's kind of a bigot and seems to take great delight in bullying Data. I'll say this right here - I like Pulaski. She's flawed more deeply than most of the TNG characters, but I think that's more because the rest of the characters are exceptional rather than the norm for people in that day and age (and, like Ro, I enjoy having a character in the midst who breaks up the harmony sometimes). Numerous other guest characters throughout the series treat Data as if he's just a machine - later in this season, Starfleet even orders him to be dismantled against his will - and Pulaski has only been on the ship for a few weeks at this point so doesn't know him. So is she behaving inappropriately? Yes - even if he doesn't have emotions to hurt, there's no reason to torment him. But I like her, for being fun to watch and for changing her opinion of Data in an organic way throughout the season, showing that even in the future, progress isn't always easy but it's possible. Don't hate! Go Kate!