First one I ever watched, like everyone else, was probably Dragonball Z.
However, I also want to mention Rurouni Kenshin. For a few years I didn't really care about anime at all. Then, there was a time when I was feeling incredibly depressed and was looking for something to occupy me and lift my spirits up a bit. I found a box containing the first season of Rurouni Kenshin and decided, since I had nothing else to do, that I'd try watching the show again after I'd just seen bits of it as a kid. After watching the first season many times, I got really into anime, and Rurouni Kenshin remains one of my favorites.