Perhaps start off with a few films as well?
Ghost in the shell
Redline
Poyo
Akira
Summer Wars
Paprika
From up on Poppy Hill
Perfect Blue
It's as ZenbaaS said, it's all a matter of taste. I had to watch a few shows to find something that I liked. I still love my shonen stuff (For the younger generation) but not as much as I used to.
I can list a few recommendations for you.
Psychological/Drama: Monster, Shiki, Eden of the east, K, The tatami galaxy (thats more of a space opera), Macross, PSYCHO PASS, Steins;Gate.
Action/Adventure: Fullmetal Alchemist, Gundam, TTGL, One piece, Naruto, Bleach, Hunter X Hunter, Yu Yu Hakusho, Magi, Katanagatari(More of a Drama/Action a lot of dialogue but an excellent show), Fate/Zero, Fate Stay/Night.
Sports: One Outs, Eyeshield 21, Haji no Ippo, Buzzer Beater, Air Gear.




enjoy the likes of Full metal Alchemist: Brotherhood and Death Note. Also if you like the more "visual" animes then Elfen Lied and Ergo Proxy are also pretty solid in my book.

