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

[SciFiBoy]

Death Note and The Melancholy of Haruhi Suzumiya would both probably be of intrest to you, i can highly reccomend both, as i gave them both 10/10.

edit: also, film wise, id reccomend The Girl Who Leapt Through Time, fits some of what you ask for, and imo, is an awesome film.