The first game I ever played was some old computer sidescroller. It was one of those games that cam with the computer I think, like SkiFree. I can't remember much about it aside from the playable character being a bit Megaman-esque. You just ran from room to room shooting things. I remember playing it for hours and never really finding an end, though I did find a room with a giant monster stuck to the wall once. I wish I could remember what it was called.
The first game I ever beat was Streets of Rage 2. It was the first game my older brother ever let me play with him and we sat there for hours until we beat it.