Computer - Radar Rat Race on the C-64, bought it in cartridge when I got the computer just to have something to start with. Last cartridge game I bought too, the rest were either on tape, later floppy disk, or typed in from a computer magazine. On the latter, I think they were trying to develop a generation of programmers with those and intentionally put bugs in them to make them not work until you fixed them. Every one.
Arcade - that's a harder one to remember, but I do know it/they would have been at a roller skating rink. Probably Asteroids as that would have gotten my attention first, or Turbo or Wizard of Wor.