The first couple anime series I watched as a kid, I never realized were anime at the time. They were stuff like Voltron, Sailor Moon, Samurai Pizza Cats, and TeknoMan. But I was a big cartoon fan (and still am) so I probably would have watched those, whether they were Japanese, American, or whatever.
The first things that I watched that I knew were anime, we're the movies that aired on SciFi back in the 90's. The two that stuck out the most were Vampire Hunter D and EYES of Mars. They are both part of my collection today, though EYES is only on VHS. I've always enjoyed animation with more grown up themes, and that's part of the reason why I enjoyed D so much.
The first anime that actually got me collecting DVDs was Descendants of Darkness. CPM actually marketed it in gay magazines and catalogs and that was where I initially read about it. I stumbled across it at On Cue (before they all became Sam Goody) and picked up the first volume. I loved the art and animation and by the end of the volume, I was hooked. I immediately went online and ordered the other DVDs of the series. DoD was also the first show that intoroduced me to the notorious "open" anime endings.
One of the biggest draws to anime now for me, is the fact that it's stuck primarily to more traditional looking animation, while most animation in the US and other places has largely moved to CGI. I don't particularly mind CGI, but I think it's bit much now that every Hollywood studio is putting films animated that way and that traditional animation has basically died on the big screen. There is a type of warmth and depth that traditional animation is able to portray that is just missing from CGI. The images in the Pixar and Dreamworks movies always seem colder than those is the older Disney, Rankin Bass, Don Bluth, Ralph Bakshi, Hanna Barbera, etc films.
Also, I love the fact that anime combines animation with more adult themes. It's why I'm such a big fan of Ralph Bakshi films, and things like Heavy Metal. They are willing to touch topics that you don't see in American animation very often, other than tags in American Dad and Family Guy. Anime series also aren't afraid to kill of major characters. I don't care how good a show is, if you go into each episode knowing that none of the cast is going to die, it takes a lot of the suspense out of whatever is happening on screen. In anime, no one is safe.