Mmmm.... more Ghibli... needs more Ghibli...
Princess Mononoke (Spirited Away fans tend to like this one, often even more for its strong action overtones)
Grave of the Fireflies (An extraordinary film that you'll probably watch once and never be able to bring yourself to watch again. Sad. Animation at its most powerful)
Castle in the Sky (A Straightforward family action-Anime classic)
Kiki's Delivery Service (Mostly only a girly-only movie. So cute it will either make you puke or fall instantly in love with it. Simple but effective and fun story)
The Cat Returns (More girlyness, just not quite as good.)
Nausicaa of the Valley of the Wind (A Fantasy Action movie quite similar to Mononoke, but predating it. The film is good, but the Manga is an instant recommendation to anyone)
and the forthcoming releases of
Pom Poko, Porco Rosso, Howl's Moving Castle, Only Yesterday, My Neighbours the Yamadas, Whispers of the Heart, My Neighbour Totoro and any others i've forgotten...
Another essential film-maker IMO is Satoshi Kon.
Perfect Blue is a Psychological Thriller that, whilst undoubtably good, doesn't have a wide enough appeal (or a low enough age certificate) to make it a "recommend this to anyone" kind of thing. On the other hand,
Millennium Actress is a beautiful film that everyone should see, and
Tokyo Godfathers is the best Christmas film i've ever seen, simply because it's the only good Christmas film i've ever seen!
Anime / Manga series you should check out. I'm going to stab in the dark and say
Fruits Basket. It's a Romantic Comedy just like Love Hina (and Utena? I've not read it), though it's aimed at girls rather than guys (less boob jokes, more prettyboys). It's really fun, sweet and quite sad in places.
Sorry, I've not got too many recommendations on the Akira side...
Ghost in the Shell is another classic of anime, but I personally thought it was pretentious, annoying and dull.
Metropolis is a nice enough Anime by Otomo (guy behind Akira), but it's not particuarly groundbreaking or brilliant if you ask me (I know quite a few people disagree with me though). All in all, don't be afraid to experiment a little. Many people find that under the umbrella of anime, they're actually interested in genres that would usually repulse them at first sight (guys liking Romantic Comedies?! Girls liking big robots :O :O :O ). There's lots of stuff out there and some of it is crap, but watch a bit more, and people will be able to give you a more specialised idea of what you're after
Edit: Just adding two wide appealing favorites of mine:
The Vision of Escaflowne (Anime): An awesome fantasy series with both a) big robot fights and b) Romantic intrigue. A rare mix that somehow works really well.
Azumanga Daioh (Anime/Manga): Amusing but very Japanese comedy about a bunch of highschool friends.