Kaleido Star. Starts out rather up-beat, but fuck the ending. Seriously, I cried like a *****.
The KEY adaptations (Clannad and Kanon), though I'd recommend reading the VNs over watching the anime. Air's adaptation is rather messy, VN isn't translated, so skip it.
Iriya no Sora, UFO no Natsu. Well, more of a personal favorite of mine, but it's only 6 episodes long, so you might as well give it a shot.
Now and Then, Here and There. Protagonist is your average shounen hero and doesn't really fit in the anime at all, but all side characters are tragic and much more developed. Themes include child soldiers, death, rape, torture (of kids), and plenty more.
Bokurano. Kids fighting in a game in a giant mech, only the game is actually real and they die when they are killed.
Infinite Ryvius. Wastes a lot of potential but is still pretty high up there.
Well then there's the whole slew of Lodoss, Code Geass, FMA, Higurashi, Death Note, etc., that aren't tragic per se but have dramatic aspects/characters. Oh and Evangelion of course, but I don't recommend it to look edgy and cool.
That's it for now, could name a few more if you want, but it really depends on if you want to watch a drama or just an anime with dramatic aspects.