Possibly around 1997-1998, I actually don't remember what was the first game I played. I either saw my cousin play Sonic 2 in a trailer park, played Earthworm Jim on Windows 95 (or was it 98..), or watched my other cousin play DKC2 on the SNES and tried to play it myself when he went to the bathroom. Memory's vague on that, so dates might not match up.
Ironically, it could also have been Duke Nukem 3D on Windows 95/98 that was my first game, we did have it and I did sometimes play it without my parent's knowledge. Sonic 3D Blast for the PC was another one (I remember the green CD), but I didnt play that one often because I was scared of the Game Over screen for some reason.
The other game I played on that computer was this weird dungeon puzzler that I had long forgotten what its name was for the last 10 years until I rediscovered it eventually though and it was this very obscure puzzle game called DROD (Deadly Rooms of Death).