Here are my favorite anime series from each genre.
Romance:
ef: A Tale of Memories
(I haven't watched a lot of romance anime)
Fantasy:
Mushishi
Mononoke
Madoka Magica
Haibane Renmei
Made in Abyss
Action:
Cowboy Bebop
Kill La Kill
Baccano!
Samurai Champloo
School:
The Tatami Galaxy
Ouran High School Host Club
The Melancholy of Haruhi Suzumiya
Kill La Kill
Revolutionary Girl Utena
Nodame Cantabile (I prefer the drama, but the anime is good too)
Psychological:
Serial Experiments Lain
Monster
Madoka Magica
Revolutionary Girl Utena