hellsing ;)

There is no such thing.

Golden Boy, and Great Teacher Onizuka

Dragon Ball Z

Avatar: The Last Airbender

Ergo Proxy, Clannad, Angel Beats, Boku wa Tomodachi ga Sukunai, Darker Than Black, Dyurara!!

Bleach <3

If you still want to see action/horror based anime, try Hellsing. It's awesome despite its age!

A good anime to watch :P

Steins;Gate is awesome!

Here are a few excellent titles that haven't been mentioned yet.
Last Exile,
Tokyo Godfathers,
Grave Of The Fireflies,
Paprika,
Gankutsuou: The Count Of Monte Cristo

Naruto, Elfien Lied, Shaman King, Pokemon, Avatar: The Last Airbender

fma,fma brotherhood,angel beats,naruto,naruto shippuden and of course dead note :D

Nodame Cantabile

Tengen Toppa Gurren-Lagann!!!! One of best animes ever!!!

The Melancholy of Haruhi Suzimiya

Bleach

Angel Beats

green green

click just go here and life will be great!

hunter X hunter 2011

POKEMON

best anime ever ;)

Tenchi Muyo!

pedo bear the animated series!
As some have said ergo proxy, darker than black, trigun, and the original hunter x hunter are great. There is also the prince of tennis!

Jormungand
http://www.youtube.com/watch?v=uDhEtUl7U-A
It's not the best anime ever or something like that, but it's pretty nice and with no digression like the huge animes as Naruto and Bleach and if you play some first person shooters, certainly you will like to see some of the guns(even vehicles if you play Battlefield games) that you use in these games in the anime.

Boku no Pico

Cowboy Bebop, Baccano, Samurai Champloo, Monster, Serial Experiments Lain, Wolf's Rain, Ghost in the Shell, Gungrave, Trigun.

Hellsing if you like violent fun/trash.

Movies:

Anything from Studio Ghibli.
Perfect Blue.
Millennium Actress.
Ghost in Shell: Innocence.

Akira is really interesting and I liked watching it, but I can't recommend it because it's mostly just a bunch of LSD trip nonsense.

EDIT: Add Haibone Renmei. One of the best.