Let's talk about which games both introduced us, or got us hooked to the franchises they're a part of.
Sonic the Hedgehog: I was introduced to it through the Game Gear games, since I didn't own a Genesis yet. The ones I remember playing a lot of were Sonic Chaos and Sonic Triple Trouble. They were good games in their own rights, but I truly got hooked after playing Sonic & Knuckles.
Super Mario Bros. : Through most of my childhood, I'd go over to my neighbor's house to watch them and my brother play Super Mario Bros 3. I owned an NES, but strangely enough there weren't any Mario games in our collection. I didn't get to play any of them until finally getting a Super Nintendo when our babysitter let us borrow Super Mario World. I'd been diehard Sonic fan until that day.
Final Fantasy: I never really understood RPGs though my early years as a gamer. They didn't look like something I had the patience for. Like a lot of people, I admittedly started to warm up to the genre after playing Final Fantasy VII.