Holy crap, RahXephon fans! I love that show.

If you want Comedy, one of the best shows to get is Azumanga Daioh.

If you want Mature Drama, there are tons to choose from. RahXephon was mentioned and is probably my favourite anime. Lots of people wrote it off as an Evangelion clone, but I feel is much more enjoyable. RahXephon made me cry.
Something more recent is Gankutsuou: The Count of Monte Cristo. It's a unique take on an already established piece of classic literature, and it's probably one of the most beautiful anime you'll see for a while.
Twelve Kingdoms is probably my 2nd favourite show, (tied with Eureka 7) but it's about 50 episodes long and ends kind of abruptly. Apparently when it was showing on TV, the project had to be cut short.

Action shows are plentiful, and one of the stand-out Action series is the Read Or Die (R.O.D.) series. It started with an OVA then went on to an excellent TV series about women who can manipulate paper. Some very creative things happen, and it's very well animated. The music is nice and jazzy as well, just how I like it.

Big Mech Action shows are a genre in and of themselves. My friend Saku can go into painstaking detail about any mech show worth watching. Gundam is the flagship of the Big Mech genre, and most of the different series are pretty good.

For Romance, one of the most well known and accessible shows is Ah! My Goddess. It's one of those classics, and was recently re-made into a new show with a different storyline. The DVDs of the new show just finished, and the 2nd season of the new show is still showing in Japan and has already been licensed. I say this show is accessible, because you can get a girl to watch it and enjoy it, and you won't be bored to tears by the romance stuff.

There are, of course, tons more. The ones I've mentioned would be good to start on and are currently available in their completed forms (Except Gankutsuou, which has one more DVD to be released next month).

Some notable up-and-comers that are starting on DVD are Eureka 7 and The Melancholy of Haruhi Suzumiya.

This forum needs an anime section.