[@The Mad Hatter] I second [@Halvtand]'s recommendation of Basilisk. Violent, great animation, and not too long. Some other stuff I recommend:

If you haven't already seen Fullmetal Alchemist Brotherhood, that's always my first recommendation for people getting into anime. It's got great animation, great story, great action, and a good English Dub. If you have a Netflix account, they have the full series there. Make sure you watch Brotherhood, and not the original, though. The original Fullmetal Alchemist is good, but brotherhood is much better. 

If you want something weird, check out Paranoia Agent (Mousou Dairinin). It's a psychological thriller that's hard to describe without giving away spoilers, but there are some stand alone episodes that are just amazing. Episode 8 happens to be one of my favorite episodes in all of anime, but only if you enjoy morbid humor. If nothing else, check out that one episode. 

My favorite anime of all time is Clannad (and season two, Clannad After story). This holds the distinction of being the only work of fiction to ever make me cry. And we're not talking just getting a little watery-eyed, we're talking full-on tears rolling down my cheeks. Season one is a high school drama, but season two actually continues on after high school, which is really nice to see in an anime.

If you want a superb story-driven anime, check out Guardian of the Ancient Spirit (Seirei no Moribito). The unique thing about this one is that it's got a female protagonist who is 1.) not a magical girl, and 2.) never succumbs to shameless fan service. This anime has an incredible story. The action is sparse, but when it's there, it's awesome. 

If you want to see the most fascinating anime villain of all time, check out Monster. Warning, this series is LONG. Like 80+ episodes. But, if you can commit, it's definitely worth it. A brain surgeon chooses to save the life of a child over the life of an important politician because the child was admitted first. This hurts his career, and several years later, the doctor finds out that this child has grown up to be a homicidal sociopath. This anime is definitely character driven and revolves around what makes the villain tick. 

Looking for sci-fi? Watch Steins;Gate. It's one of the few depictions of time travel that gets it right. No illogical time paradoxes here. It's clever, suspenseful, and has fantastic characters. Check it out. 

And, if you're just looking for people beating the shit out of each other, check out Hajime no Ippo. It's boxing. It's good. It's ongoing.

When you've exhausted these, I've got plenty more for you, but I don't wanna overwhelm you.