>>1424281Before you spoil yourself with more "new" anime (I'm talking about post-2001), I recommend watching older shows first. The animation initially looks dated, but personally I find it a lot better than the shit coming out recently.
* = watch this or you will forever be a plebeian
Cowboy Bebop*
Top wo Nerae! (aka GunBuster)*
Ghost in the Shell (1995 movie)*
Neon Genesis Evangelion*
Outlaw Star*
Berserk*
Akagi
One Piece
Spriggan
Hajime no Ippo
Hokuto no Ken (aka Fist of the North Star)
Hellsing
Trigun
Hunter x Hunter (1999 - 2001 series)
Rurouni Kenshin
and many more...
Having said that, there is a shitload of good "new" stuff to watch:
Samurai Champloo*
Tengen Toppa Gurren Lagann*
NHK ni Youkoso! (aka Welcome to the NHK!)*
Ghost in the Shell: Stand Alone Complex*
Darker Than Black
Black Lagoon
Fate/Stay Night + Fate/Zero
Soul Eater
R.O.D. the TV
Hunter x Hunter (2011 series)
Honey and Clover
And whatever the hell you decide to watch, stay away from Naruto and Bleach; you'd be doing yourself a huge favor by saving yourself hundreds of hours of shit-quality animation and disappointing story. If you find yourself swamped or backlogged, use this:
http://myanimelist.net/ (and take reviews with a heavy grain of salt).