Reccomendations? Simple.
Berserk: I've said it before, and I'll say it again. If you watch/read one anime/manga in your life, then this is it. The show starts off slow, but when it gets started, it doesn't stop. It's like reading a novel, where every chapter contributes to the plot. It knows how to be deep, philosophical, and symbolic without becoming weighed down by its own themes or becoming pretentious. It's basically the thinking mans Game of Thrones, except with better characters. It's rare to see a story that has tons of action while still remaining intellectually engaging. The Manga is more thoughtful, but the anime may be easier to get into since it skips to the best part of the story after the first episode.
Kino's Journey: A well written female protagonist in an anime? A show that's philosophical but entertaining? If you like the twilight zone then you'll love Kino's Journey. There's no overarching plot, but each episode is well thought out, and while it's no comedy there's a certain sweet dry humor to it. Basically a young woman travels from town to town, and spends 3 days learning about each place.
Berserk: What an emotional roller coaster. Drawing from philosophy, religion, classic literture, and greek tragedy, Berserk is a true masterpiece. Meticulously plotted, it is one of the best combinations of character study and epic plot I've ever seen.
Satoshi Kon: A film director I've heard nothing but good things about, if you want something more serious and psychological. If you want to find great anime you have to venture out there a little more. If all you watch is Naruto, DBZ, and Fullmetal Alchemist then you'll end up with a skewed view of anime.
Berserk: I'm an avid reader and fantasy fan, but I don't exagerate when I claim that Berserk is probably the best fantasy epic I've ever read. And I'm quite well read. The epic scale of Lord of the Rings, the character depth and moral ambiguity of Game of Thrones, and the themes of classic Greek mythology and tragedy come together to create a horror fantasy series that is so much more than the sum of its parts. It's what Game of Thrones should have been. It's what inspired Dark Souls. It's a masterpiece.