I played some games on my husband's old Apple II GS, but I won't count those. The first PC game that I was seriously addicted to was Lucasarts' X-Wing vs TIE Fighter. Since I have a TNT2 card, I can't play it anymore, but man, that was fun. Now I play X-Wing Alliance, but the online community at the Zone is so stagnant nowadays.
First commercial games: probably Starfox and Jumpman on C-64 (circa 1982).
First game on a computer that was mine: Hangman. I wrote it in basic to play on my Sinclair ZX81 with 16k of memory and a casette recorder for storage (circa 1980). Hey duckman..think you can overclock that one.
But the first games I played on any computer were "Lunar Lander" on a DEC mini and a multiplayer game called "Star Trek" on a Honeywell mainframe (both circa 1979). The DEC had a keyboard and printer only, no screen. What you typed printed out on the printer. The Honeywell had 128k and used tickertape as the mechanism for storage.