I always thought anime was 90% - what we would call - "hentai" and/or romance.
Then a year and a half ago, a few graal friends convinced me to watch Tengen Toppa Gurren Lagann -- my first anime. IMO that's a great one to start off with. The artwork is a little above CartoonNetwork-quality (so you'll be able to fully appreciate the higher quality artwork when you get to it; without being disappointed when you stumble to a lesser quality anime); and the action is so intense and the themes so compelling, its perfect to make you wanna watch more anime.
After that, I watched all of Naruto/NarutoShippuden (almost 500ep total at the time) in about one month...
Up to this point, I had only seen cartoon-style art anime. Katanagatari, Blood+, D. Gray-man all had medium quality artwork. But after watching Death Note, Darker than Black, Claymore, I really started to get into the higher-quality artwork. I also decided Action isnt the only genre I like. I also started to really get into Sci-Fi and Horror anime. Anything that a) has insane action, b) can blow my mind, or c) has a obscene amounts of blood.