There's some fantastic tips in this thread already, tbh, I find it difficult to add to them.
I'd back up recommendations for;
Legend of the Galactic Heroes - epic space opera
Cowboy Bebop - Tarantino does anime (it's not actually Tarantino, but very much that style)
Maison Ikkoku - long-running comedic romance
Last Exile - steampunk airship battles, awesome!
Miyazaki - Nausicaa, My Neighbour Totoro, Castle of Cagliostro.....actually, just watch them all!