I would have to say that it started on YouTube. Clicking around in the "Videos being watched right now..." ticker lead to my search for and eventual addiction to Anime.
Started when someone wanted me to design a Mahou Sensei Negima related site. After that, I thought, "It won't hurt if I watch a couple of subbed episodes." Got them, watched them, and the rest is history.
It wasn't long ago...the moment I heard the song: COLORS in opening one of code geass. I started watching and exploring more animes for more songs, then I got addicted to the animes timeselves. Especially Code Geass. I'm a spoiler machine for that.
Anime started with Dragonball coming to the USA. Unless your talking about comics which case thats been over here for ages. That was really the true start of the boom.
I got addicted to anime when i started to watch Toonami on Cartoon Network after school watching Sailor Moon, Dragon Ball Z. And to this day, those 2 anime's are my top 2.
I started reading Dragon ball when I was like 11 and started collecting. Then 1 year later I started to watch Yakitate! Japan together with my sister who already liked anime and it went on from there.
I also lived in Japan for 2 years when I was 9 and there I watched a lot of Yu-Gi-Oh, Pokemon, Mach GoGoGo and Atakku no. 1 :D
One of my freinds got a Fansub of AKIRA off a usenet site way back when... and it was much like crack... except Im pretty sure a drug habbit would have been cheaper.
See, before the days of anime, my friends talked me into watching Akira. They told me it was like sex but in a cartoon, or maybe it was like...anyway, they hyped it up so much for me that when we watched it, i wasn't impressed.