(New comment based on info)
I'll second the Studio Gibli movies listed elsewhere. All have been released by Disney and will be easy to find & see. All Gibli films have a feel of fantasy and or sci-fi in a really interesting and beautiful world setting, and explore deeper issues usually.
(Also add to that list Nausicaä of the Valley of the Wind, my personal fav. For you, I would personally recc Princess Mononoke and Nausicaä over Howl's moving castle and Spirited Away)
Anime or not, Gibli movies are great films for almost anyone that isn't 100% stoic and unemotional.
Now I'll mention a few things that you won't see recced elsewhere.
Scrapped Princess - Fantasy anime, with sci-fi elements. Really deep and interesting world if you watch the entire thing.
Simoun - Sci-fi, but with high tech magical airships that draw three dimensional runes in the sky to produce effects. Everyone in this world is born female, chooses gender later. While this could be categorized as a "lesbian" show, the plot, drama, and action stand on their own.
Love Stage - Comedy, pretty boys. A young boy and girl film a TV commercial. Boy falls in love with the girl, does not see her till years later. Obsesses for years. He then finds out the girl was played by/actually a boy. Damn hilarious show. My wife usually makes my straight guy friends watch it, every single one has enjoyed it.
Legend of Galactic Heroes - If you want epic hard sci-fi dealing with serious deep social and political issues, and alot of love of history. Not a light or kid's show by any measure. Very long series.