Some things to watch

Mushishi

Great example of the iyashikei genre.(TL Note: iyashikei roughly means "healing") Very soothing and slow paced, but well done and great dramatic story. Highly episodic, and I wouldn't recommend watching more than a couple episodes at a time. 24 episode first season, one OVA, and a second season beginning in April.

Natsume Yuujinchou

Another example of iyashikei. I tried not to put doubles of genres in here, but the OST to this is so good it was worth it. Highly steeped in Japanese folklore as well.

The Melancholy of Haruhi Suzumiya

While not a huge fan of the series, the art, animation, and story from the movie(The Disappearance of Haruhi Suzumiya) put this one on the list. And yes, you do need to watch the series to watch the movie, so don't be lazy and try to skip ahead. Word to the wise: watch the first season in chronological order, not airing order.

Kara no Kyoukai

A series of now 8 movies, plus an epilogue OVA, the longest of which at ~2 hours, typically around 60 minutes. Great, well choreographed action and a well done story. The 5th movie takes the cake as the best.

Eve no Jikan

A story of what may happen in the near future when androids and humans are nearly indistinguishable. Short and sweet(6 15 minute episodes, or one movie condensing and adding a few scenes), but very well done.

Tokyo Magnitude 8.0

A very well done story that uses data done by a study of what would happen if Tokyo were to be hit by an 8.0 earthquake right in the middle of Tokyo Bay. The story follows a pair of young siblings and a woman who escorts them as they travel from Odaiba as the quake hits, to their home on the outskirts of Tokyo. Oh, and you'll never be ready for the twist at the end.

Arakawa Under the Bridge

A great slapstick comedy done by SHAFT.

Katanagatari

Another with well choreographed fights, plus an eye pleasing art style. Definitely worth the watch. 12 ~45 min episodes

Sayonara Zetsubou Sensei

Another of SHAFTs comedies, though this one is highly topical on world and local issues and errs a bit on the dark side with its humor. Worth watching twice, once for the actual spoken comedy, another for the references placed throughout the scenes.

Heroic Age

A good, modern example of a space opera. No, theres no singing, but it does have a very grandiose feel. Others fitting the space opera genre: Legend of the Galactic Heroes, Macross, and many of the earlier Gundams.

Gankutsuou

Really mainly here for the intriguing art style. VERY loosely based on the Count of Montecristo.

Gunbuster/Diebuster

Yes, watch both. Gunbuster is one of Gainax's best, and Gurren Lagann is considered the spiritual successor to it, though Gunbuster and Diebuster both are significantly better.

Nodame Cantabile

A nice little romcom. At first, Nodame's squeals near ear cancer, but a few episodes in they'll become endearing.

Eden of the East

If you can get past the awful character design, its a very well written story.