Actually on that as well. Anime is quite a broad category of stuff. It developed primarily out of a lack of "real people" tv/cinema available in japan.
So they started producing animation for the purpose. Hence why there are so many "adult" themes in Anime. It's not just because the Japanese have a pervy stereotype, it actually has a basis in them targeting adult audiences.
There are some acclaimed serious anime such as Perfect Blue, Ghost in the Machine, Grave of the Fireflies, Wings of Honneamise, Tokyo Godfathers, Patlabor, Appleseed, Nausicaa of the Valley of wind(the proper release), Princess Mononoke and probably Akira.
If you then look at the slightly less serious but still very good, you get things including and like the Studio Ghibli stuff, Howl's Moving castle, Spirited away, My Neighbour Totoro, Porco Rosso, Castle of Cagliostro, Arietty
I'd probably recommend starting on some of the Movies. If you want a really serious one, go Grave of the Fireflies. It is utterly depressing, but at the same time brilliant, and shows a side of WW2 from the vision of 2 japanese children.
If you want something lightheated, Try Castle of Cagliostro, the best way to describe it would be if the japanese did a version of the Pink Panther.
Appleseed was a great post apocalyptic utopia, "mecha" movie.
Princess Mononoke, Traditional japanese mythology clashes with European influence. The Dubbed version was translated by Neil Gaiman, and has Gillian Anderson as one of the voice actors. I personally prefer subs, so I can't recall who else was in it.
I am sure there are more modern movies that were great, but these are memorable ones for me.