I don't know if it counts, but since I was a baby, my older brother had amiga with full plenty of games. Ocasionally I played on it since I was able to.
Then was an pegasus era (I still don't know, what clone it was), but it was pretty short, and ended when first PC arrived to my home. I think the first computer video game which I played many hours and still remember (and still play from time to time) was Doom 2.
First console game played: Beserk for Atari 2600 (first corridor shooter ever made)
First handheld game played: Wrestlemania Tiger games handheld (really early 90's or late 80's)
First arcade game ever played: either Galaga, Ghosts n Goblins, or Rolling Thunder (around 1984 I believe)
First PC game ever played: Blakestone (early 90's)
First owned console game played: Warhawk for PSX (wasn't allowed to own a console until I was like 15)
First owned handheld game played: Bugs Bunny Crazy Castle for Gameboy (the original pea-green screen white brick) (early 90's)
First owned PC game played: Sub-Space ( http://en.wikipedia.org/wiki/SubSpace_%28video_game%29 ) What a fun game this one was, wonder if the Euro servers are still up for this game? (1998 when I owned my first PC)
Ironically when i first played Space invaders on an Atari back in the late 80s I actually hated it. Then a few years later I got hooked on to the original Prince on the PC and never looked back since then.