Since other people are describing how they first came into anime, I suppose I shall as well.
I have to say that I'm probably best-described as more of a late-adopter. I sort of knew it existed in middle and high school, but since I spent my money on other stuff, I never really bought any of it. Cartoon Network and other such channels were only just starting up at the time. One of my high school friends has a Sailor Moon fan club, but I thought she was ludicrous. She's expressed a recent distaste or disillusionment with anime. Perhaps she's had an overload.
Another friend, this time a guy, was very much into anime early on though. He didn't much care for Sailor Moon, but instead I believe his favorite at the time was NGE. He seemed to have quite a large collection of everything, and it didn't hurt that his parents had a kickass entertainment center in their basement. It was quite impressive: a whole room devoted to just the huge TV (don't think widescreen was available back then, but I can't imagine they haven't gotten one yet) and piles of other A/V equipment including surround. I think many of my preferences developed from what I saw at his house.
In addition to large mecha animes such as NGE, I also saw some of the fantasy themed shows at his house, and that led to my first anime purchase: Record of Lodoss War. I only started purchasing anime in any quantity after I left for college. Part of that had to do with my lack of a DVD player until I had my new college PC. Once I was at college though, there was also Cartoon Network. I admit that I do like shows like DBZ, although after viewing a few episodes of the original japanese version, I really hate some of the things that Funimation did to it. I guess they figured that the fundies would be offended by Mr. Satan and Hell (not that they had anything to do with each other in the show). I still continue to watch quite a bit of anime here in law school, perhaps even more than in college.
I guess the three biggest influences on my viewing habits, in chronological order, were my friends, Cartoon Network, and the Internet (yay Bit Torrent?). What immediately caught my eye as distinguishing anime from traditional western animation was obviously the art styles. I very much liked the crisp, streamlined character designs in shows such as NGE specifically, although later on I found that I could still enjoy the shows with slightly rougher designs like Lupin the Third. I'm not sure how much of an influence that is now, since I look at western shows like Home Movies, particularly every season until the last few, and I love it, while I then look at a show like Ed, Edd, & Eddy, and positively despise it.
As someone else mentioned, many (though not all) animes also frequently have themes that aren't touched on in western animation. Looking backward, I think this is why I especially loved an American cartoon (I think) by the name of Exosquad. It was dark and gritty at times, not at all the happy go lucky utopia of other cartoons. I guess a lot of other American stuff at the time seemed to have too much "fluff."
That's about all I have to say about that.