The first game I got was Pokémon Yellow...But I don't remember anything before I played Gen III at this point... My memory is very bad.
With that said, the most significant memory I have was years ago when I played Pokémon XD. I had evolved my Eevee there into Jolteon, and I was battling Gorigan from Team Snagem I remember particularly struggling in that battle, especially against his Sealeo who was somehow cleaning out my entire team with Earthquake. For some reason, I'd thought Sealeo was ground because of that and never tried using Thunderbolt... Not that it mattered in the end, when Sealeo's Earthquake didn't even do half of Jolteon's HP somehow, and Shadow Ball did about 1/4th of Sealeo's HP. All I had to use was one Super Potion or something the entire rest of the battle, and Jolteon really just cleaned out Gorigan's entire team from then - and did it alone, too. All my other Pokémon had fainted.
Ever since then, Jolteon took over as my favorite Pokémon for being that strong and coming through like that, and being pretty darn cute while doing it.
Jolteon did it again recently when I replayed Fire Red... But that's a different story.