Like all media 90% of it is crap. Thought that is, of course, all subjective.

I'm going to disagree with dead5 here because I disliked NGE and found the whole Christian imagery bit much more off-putting than I thought it would be. Again, all personal taste. Fans of NGE usually are terrible, though. Most Anime fans are pretty terrible people to be around, unless you happen to find incredibly loud people enjoyable.

Miyazaki is very good. I would also recommend you watch the classic "good" Animes such as Cowboy Bebop, Ghost in the Shell (both the movies and the series), FLCL, and Code Geass.

I would also recommend staying away from most popular Anime/Manga. While there are a few exceptions, most of which are noted about, it is by and large garbage. Naruto and Bleach are pretty terrible, and most importantly they are terrible and LONG.

If you don't start vomitting out of your eyesockets when watching these, and feel like getting a bit deepr, I've always been a fan of NHK ni Yokoso even though the animation is terrible. The story is very good and it really did deserve a better animator. Sayonara Zetsubou Sensei is also weirdly funny if you enjoy puns. There are more but none off the top of my head.

In all likelihood you will have much the same problem with the medium as I do. The stories tend to get VERY heavy handed very quickly and character actions do not usually line up with what human beings do. That and Japanese humor seems to be a great deal more obnoxious, in the classical sense of the word, than what I find funny. Plus Anime clubs are filled with people who need to take a goddamn bath.