Like movies, you have to leaf through all the crap to find something good! Research and reviews are key, followed by sampling them for yourself to see what you think.
Series I can recommend (though others most likely have already) would be:
Death Note - it's the equivalent to a good psychological thriller movie with loads of twists and interesting characters
Naoki Urasawa's Monster - same as Death Note, but probably more complex
Baccano! - mixes crime drama, action, adventure, comedy, violence, horror and the supernatural. The story is non-linear and the events all tie-up to a violent train journey some of the characters are on - The Flying Pussyfoot.
Durarara!! - from the same creator(s) as Baccano!! but set in modern day Japan; the story is linear compared to Baccano! and it isn't as violent, but there's loads of interesting characters!
The Melancholy of Haruhi Suzumiya - a sci-fi high school comedy, which is funny, intelligent and has likeable characters
Cowboy Bebop - a true anime classic, which takes inspiration from Western movies and turns it into such an entertaining sci-fi action/adventure. The characters are cool, likeable and funny too!
Black Lagoon - (just got into this myself) it's about a band of modern day pirates/mercenaries as they take on missions and commit crimes around Southeast Asia, and how a Japanese business man - Rock - decides to join them after he's been taken hostage. His relationship with the crew, especially with Revy, who is my new favorite character and is like a more badass version of Lara Croft, is quite interesting I think. Rock and Revy seriously don't see eye-to-eye, but they begin to warm up to each other as the series progresses. This series is the equivalent to a Hollywood action movie.
Welcome to the NHK - a black comedy, which pokes fun at modern day Japanese society and even otaku culture, while following a hikikomori (recluse/shut-in) called Sato Tatsuhiro as he struggles with his social problems. He gets into all kinds of awkward and funny situations as he tries to battle his problems that he believes are all part of a conspiracy created by the NHK. This series will appeal even more if you suffer from similar social problems. Like me.
Gungrave - the equivalent to a Western Hollywood Mafia/gangster movie, but with a dose of the supernatural. This one is more character driven as it focuses on two friends, their rise to the top, how they become enemies, and how one of them comes back as an unstoppable killing machine. Currently watching this myself.
That's just some of them anyway...