In many ways, my introduction to anime is incredibly stereotypical. I used to hate it because I, like many I'm sure, thought that it was all like the crappy Saturday morning cartoons I was exposed to, such as Pokemon, Digimon, and Yu-Gi-Oh. I thought all anime was badly animated nonsense about kids with huge eyes running around summoning monsters with cards and red balls.
But then, I watched a Hayao Miyazaki film. Yep, like many, many others before me, Miyazaki kickstarted my love of anime. The first film of his I saw that really stuck with me was Spirited Away. I quite liked it, and ended up watching more films of his, ultimately watching what is now among my favorite films of all time and Miyazaki's best, Princess Mononoke.
I don't know if it was Mononoke that convinced me to try out anime TV shows, or the constant nagging of my friends who were into anime, or a combination of the two, but either way, I decided to give it a shot. The first anime TV series that I rented and saw was Fullmetal Alchemist. I'd chosen it because the title intrigued me and I'd read good things; needless to say, I was blown away.
After that, I was hooked, and still am. I love anime -- though I still hate Pokemon.