Well first, part of the hubbub is because anime (in many ways) is just a medium for telling stories.
There are all sorts of genres and styles that can be / have been used over decades to do everything
from commercials to full length feature films with anime.
Having said that, its a very "geeky" thing to get into because most of it is so different from what
westerners are used to seeing from "cartoons".
You did not mention what type of movies and TV shows you normally like to watch, so its hard to
guess a good match up for you to start with.
One "free" suggestion, if you get the cartoon network on cable, is to check out the Saturday nite
Adult swim block of animation. (it runs late but you can set your VCR to tape it and watch it later).
That gives a fairly good example of different types of stories that can be covered thru anime, on
the cheap.
If you can find a place to rent from, check out series like Cowboy Bebop, Trigun, or some of the
Gundam titles (Gundam 0083, Gundam 0080, even Gundam Wing). These are good stories that
don't get heavy on Japanese culture or philosophy, so you can enjoy them without going
"What the hell?..." thru most episodes.
Hope that helps. Just remember, you don't have to like the first thing you see, but if you don't,
look around a little before giving up, you might just find some stories made in a style or about
a subject that appeals to you more.