Even though it wasn't an Anime, my first exposure to anything Japanese was Power Rangers, which got most of it's footage from Super Sentai series. I was two or three at the time it came on, and I became obsessed with it, to the point that I recorded every single episode(I still have them too) and bought all the action figures that were available in America. About 80% of my early childhood memories are somehow related to Power Rangers, and my mother would actually sing "Go Go Power Rangers" to me as a lullaby.
When I was about five or so Fox started airing Digimon, Zoids, and Shaman King, which I really liked, but kept forgetting what time they came on and so only saw a handful of episodes.
After that my memory gets kind of hazy. I know at some point the Pokemon craze hit my school which led to me discovering Toonami, but I can't shake the feeling that I was already watching Dragon Ball Z and by that time. Which ever came first, Toonami soon began airing Gundam Wing and G Gundam and a couple other shows, which kept me watching.
It wasn't until 2003, though, that I found out that all these shows I had enjoyed up until then were connected, that they were called "anime" and that they all came from Japan. That was the year Toonami aired Rurouni Kenshin. I still remember that night. Earlier in the week I had seen the advertisement for it and it had intrigued me, though I didn't catch the date of when it was going to air. The show before it had just ended and I was about to turn off the T.V. and get in the shower when what I thought was the commercial came on again. I stuck around to catch the date, only this time, at the end, that awesome deep-voiced announcer came on and said "Toonami is proud to present Rurouni Kenshin". So I decided I'd watch a few minutes and see what it was like. And then a few minutes turned into 15. And 15 turned into 30. And when it was done I sat there in shocked silence until I realized that show had so engrossed and entertained me that I was still in the same position I was in when it first started, holding remote in the air, leaned against the chair arm, ready to get up. And then I thought "This is the greatest thing I have ever seen in my life". And the next morning I yahooed it and spent 8 hours reading all about the show and from it learning about anime, manga, Japan, and what have you.
A few months later during summer break I was up at night channel surfing using the menu guide of my dish when I noticed a show called "Inuyasha" was playing on Cartoon Network. I turned to it and with a rush of exhilaration I saw it was one of those anime thingys, like Kenshin, and that another one called Trigun was on after it, and that yet another one called Cowboy Bebop was on after it. The rest is history, although even after all these years Kenshin is still my favorite.