My first game was Sonic the Hedgehog for the Mega Drive, which I got when I was about 6. I was a big fan of the series back then, the music, level design, graphics and gameplay were all superb. The combined Sonic 3 and Knuckles was the best, I remember spending ages completing the game with every character.
The best Mega Drive game I played was Zombies Ate My Neighbours (which was just called Zombies in this country). I would spend many hours in my room playing that game.
Also, although it's not particularly old I should mention Final Fantasy VII. The first time I played it it blew me away, I never realised video games could be so expansive and deep. Of course now I'm older I've found better RPGs (Tales of Symphonia, Final Fantasy VI, etc.) but my first experience of an RPG still sticks with me.