Well, to avoid being overly negative, I'll just list the shows I'd recommend watching.
I'm going to talk about old and new shows because I have no idea what great older anime you may have missed:
(I started writing descriptions for each show but gave up when I saw the size of the library. Every show on my list should be watched, eventually. I've underlined the must-watch now titles)
Nadia: The Secret of Blue Water - a Gainax classic from the creators of Evangelion. Just don't watch episodes 23 - 36.
Patlabor: The TV Series/OVA - You won't watch this, but I'll list it anyway.
RahXephon - Because why not?
Death Note - Ever popular. Rather silly at times but highly entertaining.
Noragami - best BONES show in some time.
Gurren Lagann - An obvious choice. The super robot series.
Attack on Titan - Because you will anyway. This is one of the most flawed shows on my list but a lot (and I mean a lot) of people seem to love it anyway so who am I to judge.
Black Lagoon - 80's/90's asian action movies, if they were combined into a TV series staring mercenaries living on an island filled with criminals.
Code Breaker - An extremely competent shounen action series with super powered dudes fighting other dudes.
FMA: Brothehood or just FMA - I'd say FMA is by far the better show, but I don't have the energy to argue about it. Either one is good.
Nana
Ghost in the Shell Stand Alone Complex
Samurai Champloo
Eureka Seven
Mushi-Shi
Moribito
Wolf's Rain
Fate/Zero
Natsume
Baccano!
Revolutionary Girl Utena
Princess Tutu
Madoka
Ping Pong
Birdy The Mighty Decode
Tatami Galaxy
Vampire Pricess Miyu
Serial Experiments Lain
Silver Spoon
Bunny Drop
The Skullman
Gunslinger Girl
Gankutsuou
Kino's Journey
Haibane Renmei
Tsuritama
The Rose of Versailles
The Flowers of Evil
Macross Plus
Brother Dear Brother
Space Adventure Cobra
There. If you finish this list off (might take a year, or less, depends on your free time) or if you'd like further advice or reasoning on any of the shows listed above either find me or like minded people over in AnimeGAF in the Communities for Off-Topic or PM me with a specific request and I'll be happy to make you a custom list based off preference, genre, previously seen works etc.