My brother got a Gameboy Color shortly after they first came out. With it, he bought Legend of Zelda: Oracle of Time. I think we played through that game six or seven times apiece; I never got some of the puzzles, and had to keep passing them off to him, but that game defined the idea of a storytelling game.
Fast-forward through years of playing Sims and educational games, to when I moved out of my parents' home. My first roommate had Persona 3 on the Playstation, and he let me play it while he played obscurely complicated games on his PC. When I made the step of buying my own game system, I made sure to get a First-Gen PS3 so I could play the Persona games.
Dragon Age: Origins was the first game I completed entirely by myself, and it's helped me step further into gaming. I still consider myself a "casual" gamer, since my current roommate is a true gamer, and he's always on his system--I don't devote enough time to gaming to be a serious gamer. But I'm on my second playthrough of Dragon Age 2, trying to get all the trophies. I think I'm well on my way to becoming a serious gamer, as long as the world keeps providing me with well-written RPG games.