Just remembered another one. Post-game Ruby, I was just wandering around Hoenn, not sure for what, exactly. So I was crossing the water on Route 118 when I reached the shore, but decided to turn around for whatever reason. The moment I did, I ran into Latios for the first time. My jaw had dropped then, because it was just so random, and yet so awesome to run into a Pokémon I wasn't familiar with. To think that it probably would've taken me longer to find Latios if I had decided to just land and go on to wherever I was going.
I also swear that I caught Rayquaza on the second PokéBall I threw, but that memory's admittedly fuzzy, and I may have been playing with one of my brothers' versions because I tended to do that.
And this is a memorable moment even though it's frustrating, but my first Shiny was a Ponyta at Mt. Silver in Silver. It was in the morning before I went to school, and I wanted to catch it, but then it fled. Then over ten years later, I was playing SoulSilver and was grinding in Mt. Silver when I ran into a Shiny Rapidash. I caught it by the skin of my teeth, as it was poisoned and had a sliver of health left. I made the connection later after I caught it--even though it's more coincidental than anything.