My first was called something like Sonic Adventure Battle 2 for the gamecube.
Shortly after I bought that I got bored with it and started SSBM, which I still play today!
My first game was Need for Speed III: Hot Pursuit for the PC. It acted up funny and always froze on my horrible eMachines Windows 98, but I always loved playing it and I still play it on Windows 7 to this day.
Damn, I honestly can't remember. I think it was Pokemon....Sapphire? But it was my sister's, so I guess that doesn't count. I guess I would say LeafGreen then.