There are many excellent movie features out there, but it all depends on the taste of the individual, so there's going to be many, many different opinions as for what to watch.
Akira, a definite classic and pioneering movie, but not necessarily the best for a 1st timer.
Vampire Hunter D, great flick...if you're into the whole vampire vs human struggle thing, takes place on an alternative future 'earth'.
Same goes for Blood, The Last Vampire, another awesome flick, this one takes place in the 1940's WWII period.
Ninja Scroll, one of my favorites, if you're into the whole ninja/samurai battle thing. Also set in a fantasy earth of long ago.
Ghost In The Shell, very very good, and like ATTR said, very Matrix like with the whole human/machine interaction thing going on.
Like fantasy and magic stuff? Spirited Away was incredibly good. I can't get enough of that movie.
I'd also recommend Steamboy...it takes place on an alternative world where steam power rules. Great eye-catching animation and a very ingenious concept.
Want something a bit more realistic? Try Ghost of the Fireflies if you want to cry or get depressed. That poor kid losing his entire family in the war and all....
Blue was another good one...more of a hitchcock sort of thriller where the main character goes crazy and hallucinates, etc.
My personal opinion for a good movies/series always has and continues to be Cowboy Bebop...if you're into future earth / space-bound bounty hunters and their adventures. I love the concepts, the imagery and the soundtrack, with all the jazz and blues themes, is just too awesome.
I could sit here all day and describe soooo many good movies/and series, so my advise is to grab a few of the titles mentioned in this thread and just get started. Like everyone else, you'll like some and you'll hate some... it all depends on your taste.