My first was a game called "Captain Comic." It was a platforming game where you take on the role of Captain Comic, a guy in a space helmet and a set of overalls. You have to recover a bunch of treasures from all over the game world. Along the way you collect various tools to help--cans of 'Blastola Cola' which let you spit fireballs at enemies, a corkscrew that makes your fireballs travel in a oscillating pattern that can hit enemies crawling on the ground, a wand to teleport through hostile areas, etc. It's a sidescroller, and you move through various different boards, but it's actually pretty open. You don't have to do the levels in any particular order, and they all flow into each other. Some of the places you can go to right near the starting point are really, really hard unless you explore elsewhere, get some tools, and come back later. It was a pretty cool game--I'll have to see if I can find a version of it for my computer today!
The first game that I really, really got into, though, was either the original Sid Meier's Pirates or Sierra's Quest for Glory I: So You Want to be a Hero (back then, it was known as Hero's Quest). I still love those games!