I'm not sure why I don't love more anime more than I do. I mean, cracktastic plots! It should be tailor-made! And I love the art style, in general. A theory I came up with is that I just prefer live-action, because when I'm watching television, I focus a lot on the actor's faces and body language and voices. In anime, the voice is all I get of that. Perhaps I feel a little more disconnected from animation because of that. It's a completely different art form. It's moving art, with plot, not people moving through a plot.
My first exposure to anime was Speed Racer as a kid; I didn't really like it. My first exposure that I liked happened when my local syndicated tv station aired Star Blazers, a butchered version of Space Cruiser Yamato. I liked the novelty of it, animated science fiction and a soap opera plot; we didn't have a VCR, though, so missed episodes were missed. In those days I was still in school, and couldn't always be there to watch. Still, I considered myself a fan. I somehow missed Battle of the Planets, and have not seen it to this day.
The next thing I saw was a mashup of Captain Harlock with Queen Millenia. I liked Harlock, but not the other series. Soon thereafter, Robotech hit my tv screen--a mashup of three series this time, of which I only really liked the first one, Macross, mainly for Roy Fokker and Claudia Grant. Then Roy got killed off. [sigh]
By this point, davemerrill had introduced me to shonen anime and manga like Dr. Slump. This was in the distant past of anime fandom in the U.S., when it was referred to as "Japanimation." I learned from him how licensing happened, and where fans obtained tapes of shows, and which shows interested them. I saw a few shows, over at his house, pretty much all shonen as I recall: Dirty Pair, and episodes of Kimba the White Lion and Astro Boy (in black and white!). These were the days of the Cartoon Fantasy Organization, for those into anime history. I liked what I saw, but not enough to search it out on my own. Also, I had no VCR and no money for one, which put a damper on considerable fan activity.
Then I went off to college, and for the first time met women who liked anime, in particular someone in my freshman English class who loved not only Robotech but American cartoons like Transformers and G.I. Joe. Gradually, over the course of college, I began to get hints of shojo/josei in the world, but it was still ridiculously hard to obtain, and too expensive for my budget, which was pretty much non-existent. Also, I'd been sucked into media fandom through Blake's 7, which consumed an unholy amount of my time; my fannish activity with anime consisted of snagging first Deslok from Star Blazers, then Roy Fokker from Macross, and using them in shared universe stories.
I saw my first Miyazaki film, Nausicaa, in college. It was the butchered version, alas. I'd like to say I saw the greatness in it, but I didn't. I thought it was beautiful but incomprehensible, and did not embark on any quests as a result.
And that was it; I graduated, saw some more anime courtesy of davemerrill over the next few years, but didn't really learn more until the advent of internet fandom allowed my anime-loving friends to finally obtain shows of which they'd heard tales for so many years. And I think that should be another post.