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I'mma hit you up with some essentials:
Movies:
Akira
Ghost in the Shell
Kiki's Delivery Service
Millenium Actress
My Neighbour Totoro
Only Yesterday
Perfect Blue
Princess Arete
Redline
Tokyo Godfathers
TV:
Azumanga Daioh
Cowboy Bebop
Hunter x Hunter
K-On!
Neon Genesis Evangelion
Revolutionary Girl Utena
Shirobako
Space Dandy
The Tatami Galaxy
Tengen Toppa Gurren Lagann
All 100% me approved, all popular enough and reasonably diverse enough to give you a solid grounds on which to build your own taste, all (for TV at least) fairly recent and nothing too too long (HxH is 62 episodes but forgive me I'm a big fan). Hope you enjoy some of them. For reference my personal favourites are K-On! and Totoro.