First games I consciously remember playing are Dungeon Keeper and Daggerfall, as well as things like SkiFree and whatever else our old PC had on it.
In terms of "consoles" though, the first one I remember having was the Super Nintendo, and getting an N64 shortly after. Main thing I played on the Super Nintendo was Super Mario Bros. (mostly because we didn't buy many games for it), and the N64 was used mostly to settle arguments with games of Super Smash Bros., but we also had Majora's Mask, Mario 64, Pokemon Snap!, Pokemon Stadium, Kirby 64, and Starfox. Had access to the Sega Genesis through a friend, so I spent a lot of time playing the OG Sonic too.
But I think the one console that probably impacted my gaming preferences the most is the PS2. Not only did it let me play all the PS1 games I'd never got to play (Final Fantasy 7, 8, and 9, the Spyro trilogy, Resident Evil 1 and 2, Silent Hill 1 and 2, Legend of Dragoon, etc etc), but then I could play any sequels to those games, as well as opening the door to games I still consider my favorites. Persona 3, Fatal Frame franchise, Kingdom Hearts 2, Resident Evil 4, Devil May Cry franchise, Okami, Katamari Damacy, and so many more. The PS2's golden years aligned right with my childhood, and that combined with the PC games I still played (World of Warcraft and TF2 deserve a mention), I had plenty of variety just between the two.
Only other childhood games for me are probably Halo, Dead or Alive, and Jet Set Radio Future, but the Xbox just didn't hit the same.