Not sure if animation is your taste, but here's a few things I like.
Mary and Max [2009, Adam Elliot]: Mary, a young girl living in Australia, has no friends and her family is distant. To quell her loneliness she writes a letter to a random New Yorker, Max, who is a middle-aged man with obesity and social anxiety. The film covers their years of correspondence as Mary grows into a young woman. Mental health and relationships are a big part of the story, and it's much sadder than I thought. It really made me sit back and think.
The Illusionist [2010, Sylvain Chomet]: There's no actual dialogue, but the animation and attention to character is beautiful. The story follows a magician struggling to make his living as he moves his show to Scotland and encounters a young lady who is convinced his magic act is reality. I do warn you, the story does move very slowly but feel free to check it out even if just to look at the art.
Persepolis [2007, Marjane Satrapi]: This film is based off an autobiographical graphic novel of Satrapi's life in Iran during the 1979 Islamic Revolution. Her parents send her to a boarding school in Vienna, Austria for her safety, where she is confronted with balancing the decadent life in Europe and the casualties occurring back home. It's a dark reality interspersed with pieces of found happiness, and touches on themes of sexuality, nationalism, war, and personal identity. If you enjoy the film, I really suggest reading the novels themselves since they cover a lot more that couldn't be included.
Princess Mononoke [1997, Studio Ghibli]: Gods of the forest and a mining town clash over a fight for preservation of their kind, as a cursed young man encounters Princess Mononoke, raised by wolves. One of Ghibli's older films, and quite violent for it's genre.
Spirited Away [2001, Studio Ghibli]: A young girl finds herself trapped in the spirit world, and is forced to work in a bathhouse to find a way to release her parents and return to the human realm. This film become even more astounding once you realize that all the backgrounds were hand-painted meticulously. It's also rich with symbols and iconography, and very imaginative.
ParaNorman [2012, Laika Studios]: 11-year-old Norman has the ability to see and speak to ghosts, a trait that isolates him from his family and peers. As the only medium, he must break the curse surrounding his town before zombies overtake his life. If you're into stop motion stuff, Laika does a very good job of it. They have an interesting look to all of their work [like Coraline, and the upcoming Boxtrolls] which is a nice breath of air.
Paprika [2006, Satoshi Kon]: In the near future, Dr. Atsuko Chiba and her team have developed a device that allows them to enter the dreams as a form of psychotherapy. When the device falls into the wrong hands, dreams and reality begin to intertwine in a chase to cease the damage. This is some trippy-ass stuff, there's a ton of things in the dream world to feast your eyes on. I'd take a look at other of Kun's work like Perfect Blue and Tokyo Godfathers also.
The Iron Giant [1999, Brad Bird]: Vin Diesel, stop making me cry.