If you're down to just try to get into anime, watch Princess Mononoke. Its got lots of adult themes for an animated film, amazing action and animation.
From there, check out Howl's Moving Castle, Spirited Away, and Nausicaa.
As for shows: Trigun, Evangelion, Melancholy of Haruhi Suzumiya, claymore, fullmetal alchemist brotherhood are all good ones. If you like action/sci fi/fantasy.
Also Patlabor: The Movie
And Paprika, Ghost in the Shell 2, and Perfect Blue. These are for adults