Let's see....I remember it like it was over a decade ago...*screen ripple into flashback*
I was looking around the closet in the dining room for something to eat (this wasn't the pantry, but my parents had a habit of hiding sweets in there sometimes, and I was desperate. Instead, I pull out a Sega Genesis cartrige box, Mortal Kombat. Eventually convincing my Dad to plug it in and let us try it (myself and my younger brother)...well, let's just say Sub Zero was my favorite. Anyway, the first game I really enjoyed that came after that (like, MK was okay, but REALLY enjoyed) was the gem of a game Rocket Knight Adventure, staring an armored possum with a jet pack and swords, fighting giant deformed robots driven by pig monsters that were clearly the time traveling spawn of Ganon (or not. Not sure which came out first, and I actually suspect it was Zelda, so maybe not...), trying to rescue the princess, beat a renegade Rocket Knight and stop the world by being destroyed by...well, I won't spoil it. Definitly a great game. Fantastic music for the time too.
*end flashback*
So yeah. The first time it became clear that I was really a gamer though, was the eventual purchase of an N64 and getting a copy of the Legend of Zelda, Occarina of time. First big game I ever played and have loved much of the series since.