Don't really read that much in the way of manga, but I've steadily been putting some anime way under my belt over the years.
Psycho-Pass - A really effective blend of detective fiction with dystopian science-fiction - it does both very well, and the synergy they create is something you should check out if you enjoy either genre.
Baccano! - With a narrative that follows a host of different characters with a very disjointed chronology, you probably won't find your feet until the second episode, but it's fitting for a show that thrives on chaos as a host of different characters and factions struggle to get what they want on an ill-fated train ride in Prohibition-era America.
Gankutsuou: The Count of Monte Cristo - With a very distinct art style and set in the year 5000, Gankutsuou takes the classic story of revenge and changes it to a mystery, showing events from the perspective of the young Alfonse, rather than the Count. Excellent all-round, and if you've never seen/read the Count of Monte Cristo before, you should DEFINITELY check this out.
Mushi-shi - A calm and beautiful show, Mushi-shi follows the journeys of Ginko as he helps people with mushi, spiritual micro/macro-organisms that form an important part of the ecosystem. Each episode is like a well-written folk story, it's well worth a watch.
Hellsing Ultimate - If you can enjoy stupidly over-the-top but really cool violence, give this a go. Vampires. Nazis. Nazi Vampires. Not the height of intellect, but I had a blast watching this.
Steins;Gate - A bunch of misfits playing make-believe at being mad scientists accidentally create a time machine. A really good show that starts off with a cool premise and interesting characters, and explores and develops them all to the fullest.
Fullmetal Alchemist: Brotherhood, Attack on Titan, Code Geass: Fighting, Intrigue, High Stakes, entertaining characters, plot twists - all good shows, all somewhat similar.
Also, while not actually anime, Avatar the Last Airbender and Legend of Korra are some of the best things I've ever seen, hands down. If you're going to watch one thing on this list, it should be Avatar.
Smolderin said:
Zankyou no Terror - An interesting show I picked up as a recommendation from GRArkada (youtube personality) about 2 young men committing acts of terrorism in Japan. Very moody, beautifully animated, and gorgeous sound design (done by Yoko Kanno...ya, Ghost in the Shell Yoko Kanno). I haven't heard much of this show around the internet so I am guessing this is flying under alot of people's radar which is a shame because it is turning out to be an excellent anime.
I've kept my eye on Zankyou no Terror and Aldnoah.Zero as soon as I heard about them, both have excellent pedigrees - Aldnoah.Zero's being written by Gen Urobuchi, who also wrote Fate/Zero, Psycho-Pass, and Puella Magi Madoka Magica, and I'm really enjoying it so far. Zankyou no Terror's got Yoko Kanno handling sound and music, but even more impressively, it's being directed by Shinichiro Watanabe of Cowboy Bebop and Samurai Champloo, and so far, he's doing a great job, the show has this beautiful bleakness about it. I'm not surprised it's not being talked about all that much - it's a brand new anime, only 5 episodes have aired, and it's got a good deal of slow-burning mystery about it. Once they get far enough in that the mystery starts to unfold, it'll generate some noise.