>>555214start with Death Note then Cowboy Bebop and Samurai Champloo. also check out this presentation it just a very nice overview of japanese animation and some of the top studios, animators/directors.
https://www.youtube.com/watch?v=5ikHAMDbXlc&list=PLuPNSyztkHPqV-M4ePSmN_2BLhONu0WLg I would also recommend watching some of the 'edgy' stuff at first, series like Soul Eater, Claymore, Hellsing Ultimate, Trigun and Elfen Lied and maybe Code Geass are way more fun to watch at the beginning than when you watch more.
there's also quite a few that are very popular that pop everywhere so you might want to check them out too, here's some
The Melancholy of Haruhi Suzumiya
Haganai
Oreimo
Berserk
FLCL
Hunter X Hunter
K-On!
Kill la Kill
Gurren Lagann
Chunnibiyou
Naruto+Shippuden
Serial Experiments Lain
