First and foremost, I strongly suggest that you become a Troper. TvTropes catalogs the elements that go into a story and cross-references them with stories. They have a large section devoted to anime.
I always recommend Mahoraba to newbies, even if it is comedy.
“Case Closed” is a murder mystery series, and while it is often light-hearted, there is a lot of drama. Because people are murdered, usually in dramatic ways and for dramatic reasons.
“Clannad” has some drama, as does “Kanon,” though the latter also has some Shinto-flavored supernatural elements.
“Rurouni Kenshin” is a historical action/drama, with half-decent scientific backing for the bizarre stuff and a devoted fanbase. The author is something of an X-Men fan, though it doesn’t show in his art.
“Honey and Clover” focuses on the lives of a handful of college students. Can’t really say more, since I’ve only seen a couple of episodes.
“Genkishen” is said to be good; I have yet to watch it.
Senjou no Valkyria: Gallian Chronicles is set in an alternate-universe 1930s Europe, in the middle of a resource war. I haven’t actually seen it, but the setting is intriguing.
Perhaps less your speed is Time of Eve, set in the near future where humanoid robots are ubiquitous. It starts when the protagonist notices one day that sometimes, his gynoid goes someplace entirely on its own, and refuses to acknowledge it. Drama, light sci-fi, and a recurring nod to Isaac Asimov.
Miyazaki’s “Castle In The Sky” has some drama, some fanta-science, and a steampunk setting. It’s also Miyazaki, which guarantees a certain level of quality.
On the manga side, anything by Komi Naoshi. Like Mahoraba, I always recommend him to newbies. I don’t think of his work (oneshots, mostly) as being especially dramatic, but they are quite good. With the exception of ‘Apple’, which is just plain weird. The settings tend to be realistic with a tendency to edge slightly into the fantastic.
There is a lot of anime out there, but most shows boil down to the same few stories, regardless of trappings. I suggest that you experiment with the genres; you’d be surprised with what you end up liking.
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I just read the TvTropes for School Days, and wonder how anybody could like a show full of infidelity and ending in grisly murder. O_O