I remember finding and catching my first shiny. FireRed was the game.
I was training in that patch of grass, right before Cerulean City. A Spearow jumped out, and I remember staring blankly at my screen for a minute. Cuz, I'd been grinding in that patch of grass for a while now, so I was used to just KO'ing the opponent Pokemon. But these warning bells started going off in my head and I remember this mantra of "catch it catch it catch it caTCH IT CATCHITCATCHIT."
So I broke out of my stupor and REALLY looked at the screen, then went, "Holy cheezEITS IT'S A SHINY!"
The next couple minutes were spent trying to do as little damage as possible at a time because I was nearing too high a level to be training in that patch of grass. Plus trying to keep it asleep, but the thing kept waking up. Eventually I caught it and I sent a pic to all my friends. Geeked out a bit, then went on my way in game.
I was stupidly happy and excited. But, I had no need for a Spearow on my team, so I never used that Shiny.