On my first PC, Minesweeper, very shortly followed by Wolf. Whilst at Uni, that expanded to Doom, Descent, Grand Prix (Geoff Crammond's), lemmings and Risk (well at least on my corridor).
Having said that, I did have an Amiga before that, with a host of games, though being fair, I first bought it for F16 Falcon.
Prior to that, I was a Speccy user, with the rubber keyboard. A friend of mine had a BBC with Elite and Revs on it. They were awesome at the time.
First ever computer game I saw was on a mainframe. A Star Trek turn based ship combat game. Unsurprisingly rather basic. That was probably in the mid-70s, on an ICL unit. I seem to remember it using winchester removable storage disks, which were roughly the size of a baking tray and probably 2-3 inches thick. Incredibly, I've found an image of what it looked like: