don't worry Taiyo. You don't have to be 'old school' just because one generation pushed you further to be a 'gamer' and then you start to focus more attention on the games before that era.
Gaming for me started all the way from NES, but I sucked as [parasitic bomb], so a summary of me back in the day is, "lololol Mario~ jump jump jum...[tornado fang]...WAH!" *turn off*. I slowly did got better, but only played it bits and bits while I watched my brother beat a game beginning to end (even the times where he got a tad frustrated by some hard segments).
So yeah, I think the first game I actually beated by myself would have to be Super Mario RPG (I think...) and man I felt so good for that, but, I still wasn't super into gaming that i'd read articles of E3 and such. Only 2 games truly got me into gaming and that FF7 and Pokemon. Those two gave me the final push and Pokemon also introduced me to emulation because me and my family don't have the means to buy 1 console per generation, so I got my Psx late (I even got my PS2 and DC late, but we don't regret so.). So emulation was quite a big thing for me back in...whenever Pokemon Red and Blue came out in the NA.
So don't feel out of place Taiyo.