I'm no anime expert, but i'll recommend what i've liked so far.
Attack on Titan- I think this fits what you are looking for, lots of action and humans are pretty much at war with the titans.
Clannad- It's one of, if not, the most emotional anime I have seen so far (you might wanna get yourself a box of tissues, especially once you get to After Story). it is also pretty funny as well, but that depends on your taste in comedy. I can even go as far as saying that this anime changed the way I view life.
The Melancholy of Haruhi Suzumiya- Something about the supernatural in this anime made it really interesting. It does contain some fan service though, so I don't know what you might think of that. If you do decide to watch it, when you get to the endless eight arc just watch the first, second and last episode of it. I also recommend you watch the movie, It's really good, but you need to watch the series first to fully understand it.
Hajime no Ippo- This is a boxing anime, I don't really enjoy sports myself, but I really enjoyed this anime. It's not all 100% boxing, it also has its (pretty dirty) comedy and romance. The first season is pretty long at 70+ episodes, but it's well worth it.
These might not be your taste, but it doesn't hurt to try something you normally wouldn't.
