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After 7 years of living n Japan, I am JUST NOW getting into Anime.
It just never piqued my interest until my ex-girlfriend (who I am still friends with)
told me a couple months ago to give Evangelion a try. She said it would be something
I would like since she knows I only watch dark and depressing movies. So I gave it a shot.
I watched the entire series, the movies, and the first two parts of the new movies.
I was hooked and I needed more, so I bought all of the manga.
I read all of the manga. The main story and the side-story manga.
For those who saw the entire series, the side-story expands on the
dream Shinji had when everybody was normal. His mom was still alive and there is no NERV,
angels, or Evas. So the side story is an alternate reality where there are no battles.
I bought most of the older Eva games for Saturn, PS1, PS2, and DS.
I even got Puchi Eva, which is Evangelion for little kids.
I have milked this franchise dry and I need a new one.

I am not into stuff like Dragon Ball or Vampires. So I am looking for suggestions of anime with deep storylines. I tried getting into Death Note by reading the first 5 manga, but the whole premise just seemed kind of lame to me.
People tell me to give Akira and Ghost in a Shell a shot. So I will.
Are there any others you suggest?

Thanks in advance.

I've been out of the anime loop for a while, but here are some ideas in no particular order from things I have watched off the top of my head.

Rahxephon
Argentosoma
Gungrave
Planetes
Cowboy Bebop
Fullmetal Alchemist
Last Exile
Bubblegum Crisis & Bubblegum Crash (the old ones, I haven't seen the 2040 stuff)
Miyazaki movies are a sure bet
Wings of Honneamise/Royal Space Force (this was Gainax's first movie way before Eva)

I will second cowboy bebop (the movie was great too).

princess mononoke

Since you like dark and depressing movies, try these:

Grave of the Fireflies
Wings of Honneamise
Battle Angel
Spirited Away
Ghost in the Shell

I don't really like a lot of anime, but I do like dark and depressing movies, so hopefully these will fit what you are looking for.

Berserk (Even if it isn't as great as the manga, the anime is still worth watching.)
Paprika (You should give all of Satoshi Kon's stuff a try, actually.)
... that's about all that springs to mind, that no one else has mentioned. And they are substantially dark with plenty of depth.

Alot of good stuffs already been mentioned, but i'd also add these movies;

Freedom
Paprika
Tekkonkinkreet

And as for series;

Mononoke (beautiful, unique, uncomprimising series about an excorcist in fuedal Japan - very odd and innovative art style with wonderful direction and some highly entertaining stories based on traditional Japanese ghost tales).

Also, Wings of Honneamise is a wonderful film, definetly check it out.

A lot of solid suggestions in here. I'd also recommend Bakemonogatari.

I wanna add fooly cooly even though its not really dark or depressing, but it is a very funny, surreal, and extremely entertaining 6 episode series. Maybe something alittle light hearted to watch after you get through with the darker shows

It was also made by Gainax (who made eva)

Denno Coil: Circle of Children
(although it's not depressing or dark)

Kaiba
(which is one of the best series I've seen in YEARS)

I'm going to recommend Mobile Suit Gundam because it is a classic. Sure there is a lot to make fun of in it but it's also pretty mature. I would just watch the original series and see how I like it from there.

Anything by Hayao Miyazaki. He had an anime way back in the day called Future Boy Conan that is really charming and the rest of his work is so creative that it creates a wonderful experience. Most of the time it's not that dark so sorry about that.

Anything by Osamu Tezuka. The godfather of manga and anime. A lot of his stuff is targeted at children but the manga is so sophisticated that it makes manga an art form.

I'm also going to add in Cowboy Bebop, Paranoia Agent, FLCL, Lupin III, Akira, Samurai Champloo, and Ghost in the Shell.