FoxbodyMustang: The Bible Black wasn't exactly recommended to me by anyone. I was, and still am, completely new to anime and the title sounded like something I would be interested in so I downloaded it. I love dark and scary. I didn't know the difference and had never even heard of hentai and was just thrown off. That is not to say if a female friend threw it in while I was there that I would just turn it off and walk away, if she wants to throw in a porn I'm game. lol I just know that the stuff that was being shown on TV was just not appealing to me. It's possible anime just may not be for me at all. But I figure I'd give it another chance. Maybe this time take some recommendations instead of running blind. I do appreciate the advice.
I like zombie movies and apocalypse related stuff too if there is anything on that topic also.
When it comes to Horror Anime I'd recommend:
*) High School of the Dead
*) Gakkou Gurashi
*) Parasyte: The Maxim
*) Blue Exorcist
*) Elfen Lied
*) Another
*) Shiki
*) Seraph of the End
*) Vampire Hunter D and Vampire Hunter D: Bloodlust
*) Hellsing and Hellsing Ultimate
*) Attack on Titan
*) Corpse Party
Warning! Some of the above are very gory and not suited for children. They are generally rated 16+ or even 18+ in some countries. I'm pretty sure you will like all of the above and can watch them in no particular order. I recommend starting with Corpse Party and Elfen Lied though since they are relatively short.
The two When they cry series (Higurashi no Naku Koro ni (aka When the cicadas cry) and Umineko no Naku Koro ni (aka When the seagulls cry) are also very good. However, they are very difficult to follow.
They are anime adaptations of games (or rather visual novels) and every few episodes the story is restarted and sligthly different behaviour of the main characters causes different things to happen (e.g. different people becoming paranoid and killing everyone in sight or finding out more or less about the cause of the paranoia in Higurashi) until the viewer knows the whole story. So they are probably not that great for an anime beginner.
When you have seen several other anime and you like movies where the viewer never has the whole story but only sees fractions of it (possibly even time shifted) like in Memento or Pulp Ficition then you should definitly watch them, too.
On top of that I recommend:
*) Death Note
*) Fullmetal Alchemist Brotherhood
*) Black Lagoon
*) Psycho Pass
*) Avatar: The last Airbender
Avatar is great and should be easy to come by because it is technically not an anime. It was made in the US. It does copy the anime style almost religiously though and is considered to be one of the best animated shows ever.