That varies a huge amount. It’s not a genre, just a medium, and like any medium there are a wide variety of genres made with it. Studio Ghibli tends to make surprisingly thought provoking children’s movies, often without real villains. Cowboy Bebop is a hard sci-fi show in a constructed world with a jazzy sound track, and was probably the inspiration for Firefly. Ghost in the Shell is the ancestor of all modern cyberpunk, but with the quirk of being from the (still sympathetic) perspective of government counter-terrorism agents. GitS also tends to be heavily philosophical. You’ve got slice-of-life feel-good shows like Azumanga Daioh and K-On. There are children’s and teen’s shows about saving the world, and brutal deconstructions of those shows aimed at adults; Sailer Moon -> Madoka, Getter Robo -> Gundam -> Evangelion (second round of deconstruction went hard), etc. Then you have the genre-busting quirky stuff, like Haruhi Suzumiya (which is fairly sane) or Kill la Kill (which is decidedly not).
There’s probably something out there in that space that you’d love, but it’s a good idea to start from a genre you already like and look at that rather than just going with whatever show is big right now. The stuff that got really big in the US, like Naruto or One Piece, isn’t generally my thing, and that’s probably what you’ve run into. If you want to try a movie, check out Spirited Away, Princess Mononoke, Akira, and the first Ghost in the Shell movie. If you want to jump straight to a series, Cowboy Bebop, Haruhi Suzumiya, Last Exile, Death Note, Ghost in the Shell: Stand Alone Complex, Darker than Black, Serial Experiments Lain, Attack on Titan, and Spice & Wolf are all good without being too out-there.