Skipping consoles (Coleco Telstar, Atari 2600, Intellivision, Odyssey II) and the Commodore 64. The first computer game that I really recall playing was an early Ultima on a Macintosh in high school. The first strictly PC game I recall playing was actually...well, two games at roughly the same time: Steve Jackson Games' Ogre and some kind of World War II flight simulator that let me fly an SBD Dauntless in the Battle of Midway. That was in 1988 or 1989, and I didn't get back to PC games until 1994, when I picked up my own PC (486) and picked up a slew of games: X-Wing, TIE Fighter, Doom, Doom II, Heretic, Flying Carpet, Sim City, Star Control II, Master of Orion, Civilization 2, Descent, Rise of the Triad, and so on.
I guess of all of those, I played Doom first. I still regret not purchasing The Elder Scrolls: Arena, X-Com: UFO Defense or Master of Magic when I saw them on the shelves - I have played those since, but it's not the same.
I'm also missing numerous other games. I recall that my cousin (who was living with me, and also had his own PC) was constantly playing my games, too... Fun times.