Thomas the Tank Engine for the SNES. No, seriously, that was the first game I played as a kid.
In reality though, while I had been exposed to video games and played them all throughout the NES and SNES lifespans, the first game that really made me sit down and tell myself, "Hey, these video games can be good. You should ruin the rest of your life with them play more of these and pay attention to what they're telling you," was Ocarina of Time. Yeah, yeah, cliche.
But I was a kid. I had fun with it. And the game is still incredibly good, even to this day. I loved the puzzles, loved the action and loved the visuals. (N64 was still new) Everything about OoT engrossed me and I can confidently say that it was because of that game that I went out and proceeded to play every game that I could get my hands on see if anything else could give me that awestruck feeling again.
And I found out a lot of games did.