I remember picking Squirtle in Blue Version because I thought you were supposed to (the Blastoise art, of course).
Nicknaming said Squirtle 'Bubble', only to find out that it was the name of an attack. Bubble was my first Pokemon ever. Eventually, as a Blastoise, I found the name unfitting, and renamed it 'Icemonster'.
Bumping into the walls in Dark Cave for hours on end. D:
In a different Blue file (same game, though. Still have the cartridge as a matter of fact), I named my character and EVERY Pokemon that I caught BLUE. I thought it would be cool, until my cousin asked how I would distinguish them in Bill's PC. After looking, I simply started my game over (having multiple boxes full of Pokemon at this point). XD
Being completely amazed when I found out you could change the color of the game by pressing random buttons on the 'GameBoy' screen. I toyed with it for hours XD
Playing Blue Version on Pokemon Stadium. That was AWESOME. :O
Not really one of the mainstream games, but it was the first Pokemon game I've EVER played. In Pokemon Snap, in the tutorial, I remember getting a photo of a Pidgey, and running into the kitchen to my mom yelling "Mom, mom, I caught a Piggy!" Later did I realize that I didn't catch it, only got a picture of it. I was distraught. :O
EDIT: Ugh, this made me REALLY want to play Blue Version.