I've never watched an Anime, but I want to start. Recommend me a great one.

[emilie autumn]

just watch lain, is pretty much all that you described, no superpowers, superb plot(and confusing as hell) plot and its only 13 episodes long, unlike those 300+ shonen with no goal ni the story. you can also go for evangelion, but unless you can stand screaming traumatized children, don't.