I recall the first game that really sold me on where computer-based gaming could go was Wing Commander. The integration of "shoot 'em up" and storyline (including alternative storylines depending on how you did) just had me amazed at the time. I actually have it on DOSbox and still play it from time to time, including the use of an old-fashioned joystick controller.
I still remember that one. It was pretty amazing at the time. (And, though it would never win any awards, the movie wasn't terrible. It was bad, yes, but entertaining. I particularly like that Hercule Poirot was the captain of the starship.)
I was already hooked on gaming before then, though. Legend of Zelda had me a few years before Wing Commander.