If you liked the first 2 (although the Englishized versions are crappy imo), I suggest you go see more of Ghibli/Miyazaki work. The last two movies are qualitatively a-ok, but they lacked a certain grittiness for me and seemed to have somewhat superficial stories. The fairy-tale passed me by, instead of dragging me in.

Great Miyazaki anime:
- Kaze no Tani no Naushika? (Nausicaa & the valley of the wind)
- Mononoke Hime (Princess Mononoke) (I'm sure you know this one )
- Porco Rosso
- Neko no ongaeshi (The cat returns)

Non-Miyazaki Ghibli productions
- Haturo no haka (Grave/tombstone of the fireflies) (Although this is no light-hearted fairy-tale, but a realistic drama -- a very good one)

Ofcourse, but I actually feel guilty recommending it, since I can't imagine someone here doesn't know it yet, Akira is not to be missed.

I've been out of it a while, but if others pop to mind, I'll post em here :]

I unfortunately have not kept up with various anime shows.

However, I can make a few, albeit older suggestions.

Puni Puni Poemi - It's a very short-lived series that's chock-full of randomness and parodies of other shows.  The english dub is especially fantastic.

Excel Saga - Excel Saga is like Puni Puni Poemi Lite.  It's a full-on anime series with just enough random humor to drive you insane and keep you watching it.

Trigun - I believe the anime runs 26 episodes long, but it has some great humor and it also has some great drama.  The english dub is also pretty damn good.

Great Teacher Onizuka - The story of one teacher at a troubled high-school in Japan.  There's lots of great humor in this one, but there's also some great drama too.

Now on the dramatic/romance side of things...

Fushigi Yugi - Spans many episodes and many series, but I really enjoyed what I saw of it.  Essentially, a young girl gets pulled into another world by a magic book, and she is seen as the savior by one of four warring nations.

Love Hina - The obligatory romance anime.  It's quite funny too if a bit dragged out.

Great series that were never finished...

Dragon Half - It's damn funny and the dub is pretty good here too.

Bastard!! - Stereo-typical VA's but a lot of humor and a lot of good action.  Dark Schneider is a pimp.

Obligatory violent/thriller anime...

Vampire Hunter D: Bloodlust - The original Vampire Hunter D was also decent, but I just like the animation quality in this.  Both movies are based off a graphic novel illustrated by Yoshitaka Amano, the lead character designer for FF1 - FF6 just about.

Perfect Blue - Suspense/thriller.  By the end of watching this movie, you'll hate the line, "Excuse me, who are you?"

Stuff I recommend you stay away from...

Akira - The majority of the movie involves some guy screaming "Akira" which causes a lot of shit to explode and many people to die.  If you must watch it, don't expect a sensical plot.  Do expect lots of impressive hand-drawn animation and orange blood however.

Ninja Scroll - The whole reason why this damn movie is so popular is because of how much sex and violence is in the film.  It's not a hentai per se, but you will see a full-on sex scene as well as reptiles emerging from a certain part of the female anatomy.  The plot is decent, but the dialogue is horrible.

Ghost in the Shell - Impressive animation, lots of explosions and violence, non-sensical plot, shitty ending.

Edit: Oh also, if you like giant robots, I recommend the Gundam series.  I like a lot of the older stuff that's based in the same universe.  (Mobile Suit Gundam, Z, ZZ, and so on.)

Ninja scroll isnt any where near hentai its just a rated R cartoon. Back in my day all anime was like that. Full of sex and violence and jiggly bosoms. Since no one here seems able to recomend anything more than trendy teenybopper crap ill recomend a few classics.

Dominion tank police: police drive around in tanks. A classic from the days when watching anime could get you beaten up.

The Slayers: a fantasy setting with mulitiple jokes about the lead characters small bosom.

Project A-ko : a 16 year old red headed schoolgirl named A-ko who has super human strength for no Apparent  reason ( thought theres kind of a subtle joke about her being the daughter of superman and wonderwoman. Her dad's  name is clark and reads the daily planet and her mom's name is diana.) fights   the robotic minions the evil genius B-ko. Why do they fight? Woman issues i imagine. This was the first anime i saw in the early 90s on the sci-fi channel and it began my frequently frustrating quests for anime entertainment. Anime didnt become commonly available untill around the mid 90s and even then a VHS could put you back 40 bucks. Im not kidding.

Wicked City: two "black guards" defend an ancient lecherous diplomat to ensure the success of a peace treaty between the human world and a dark mysterious alternate universe called the Black world. This was directed by the same guy that did ninja scroll and it has alot of the same traits. Dont watch with your kids.

i suggest AMV hell if you've got the patience for bittorent.