~Rogue~

Watch the ones I suggested if you want feels. Both of them made me cry, which no other form of media have managed to do. Start with Madoka Magica

Darker Than Black gets pretty emotional at times.

But yeah, Evangelion can give quite a lot of feels.

NG:E, as others have stated, is good (personal favourite), as is Bokurano: Ours, which is a little like like NG:E, except about a million times more sad. Less self-destruction but a lot more crying on your part.