The Revolutionary Girl Utena movie is a retelling of the series. I think it would make much more sense if you watch the series first. (Not that I would know, I watched the first 3 or 4 episodes, read what manga translations I could find, and then watched the movie)
But it’s really fantastic to trip out unsuspecting friends’ minds with.
I second the vote to see the TV series first, or at least know what happens in it. I have no idea what the movie’d be like without that… just random nonsense, I guess. The series is just as weird but is much easier to understand.
I liked the Utena movie more than the Evangelion movies, though.
Oh yeah, I trained her well. 
OOC, did you watch a fansub, HK DVD, or did they sneak a US release past me?
For me, the movie is Spirited Away by a landslide.
The TV race is a little closer with Rurouni Kenshin holding the #1 spot, but with Eva, Fushigi Yugi, and FLCL not too far behind.
I’m grateful I had a friend to push Evangalion on me. Every time he came for a visit, he brought his DVD set with him, and we’d spend the morning and afternoon post-partying hanging out on the couch working our way through the series.
I found the whole thing rather enthralling. Also, imho, I think I got more out of the series than some of the other people he’s exposed to it because I never expected it all to cinch together and make perfect sense. I drew some conclusions, but just generally enjoyed being along for the ride.
I’ve only seen a few other things, such as Akira, Princess Mononoke, Vampire Hunter D, Demon City Shinjuku, Ghost in the Shell and Kimagure Orange Road–all of which I’ve enjoyed and would recommend.
La Blue Girl series, by far.
My favorite series is Mobile Police Patlabor. I admit the premise sounds a bit silly (a “Special Vehicles Division” of the police that uses giant manned robots), but the stories focus more on the characters than the technology - some episodes have no action scenes at all. The episodes are consistent and have strong story arcs. A bit like Buffy in many respects.
My favorite animated movie is Nausicaa (aka Warriors of the Wind). It’s one of the few movies that made me cry.
Spirited Away, definitely. I was getting a bit tired of the whole cyber-mythos hyper-masculinity anime themebase.
God, I hope the Academy voters actually watch it before casting their ballots…
Last night we watched Spriggan. The plot was a little far-fetched IMHO (I mean, even for anime), and it was pretty bloody, but the visuals were stunning. The movie was losing me until the chase scene through a Turkish market place. Whoa, then I had to sit up and pay attention. Fluid movement, interesting camera angles, detailed backgrounds with figures that actually moved, it was amazing, way above average. This may not be one of my favorite movies, but it’s definitely worth watching.
(For Akira fans, Otomo was a “general supervisor” and producer on this film. It may remind you somewhat of Akira, especially when the little blue government-experimented-on kid with super mental powers shows up half-way through.)
Bakuen Campus Guardess.
… not that most people have ever heard of it, much less actually seen it… :X
Other than that, I’m currently watching Koukaku Kidoutai (The Ghost in the Shell) TV series and finishing up Witch Hunter Robin.
My older favorites would be Macross Plus (I normally don’t like mecha, but it was just so awesome in all regards) and Cowboy Bebop.
I honestly don’t believe no one mentioned Berserk as favorite series, but hey…maybe I’m just strange like that. It’s definitely my favorite series, though the manga is much better. 
Anime movie, however…Grave of the Fireflies, definitely.