With all things, in order to appreciate it--to KNOW about it--you're gonna have to expose yourself to it as much as possible. It doesn't matter what, just watch it. I'm a fan of the original, if you can help it, always go for the original version. Uncut, unedited, and much, MUCH better voice acting. That said, let me introduce you to
www.crunchyroll.com
All of the animes suggested above are excellent. I'll just throw in
Death Note and
Hikaru No Go to the bunch.
Death Note is a very, very stylish anime (and manga) with a Gothic/gloomy atmosphere. Extremely suspenseful and fantastic character development.
Hikaru No Go is about a boy possessed by an ancient ghost who loves to play Go and eventually, he learns to enjoy Go and is a very good at it, too. Again, great animation and story telling.
The thing about the above is that they're very tamed compare to animes (they actually wear normal clothes
). No fighting, no big monster or spaceship, treasures, or any thing like that. Just some good ol' fashion story telling, kick ass character development, animation, voice acting, etc...