The first anime I ever watched was SDF Macross, which I watched with my mom's boyfriend (who was a total nerd) when I was like 5. I liked it, but it was just another cartoon to me, ya know? It wasn't until probably about middle school that I started to watch anime regularly, and that was because of Toonami. The first anime I really fell in love with though (and which made me a fan of anime for life) was Gundam Wing.
If your going to watch anime, start it out right. Avoid Bleach, Inu Yasha, Naruto, and stuff like that, unless your like 10 years old, then go for it. Start out with something good, like Kino no Tabi (adventure), Spice and Wolf (romance), Black Lagoon (action), or Jigoku Shoujo (horror).
That's a pretty broad spectrum there, you should find something you enjoy among that list.