Based on your profile pic and name I'm going to assume you're a teenager and relatively new to this whole anime thing. You probably want some shounen trash like Tokyo Ghoul. I would recommend Deadman Wonderland or Elfin Lied.
Outside of typical shounen I recommend Usagi Drop or Angel Beats maybe Serial Experiments Lain. Baccano/Durarara are always good.
If you want something like Death Note I would say Psyco Pass, or the more psychological Ergo Proxy.

For watching it, unless you have crunchyroll premium, I recommend learning to torrent and using nyaa, or streaming on kissanime.