I'd always advise reading Mangas of popular Animes as well, since they are usually better, and dubbing can often be a terrible bane for English crossovers. I would have to agree that a great majority of the anime I've seen is convoluted in its storytelling (although this tends to be a Japanese thing in general, I've noticed) and juvenile in its content, but some of it is quite good.
I don't need to mirror what's been said about Akira, since my experience was similar to people like Jack's. I was a kid when I watched it, and it totally blew me away. Not just because it was so hard and violent in a way I hadn't seen in a cartoon, but because there was actually a story and world that was so unique to me. Consequently, I haven't seen one since that really resonated with me on that level as far as "film animes" are concerned, and while I enjoy movies like Ninja Scroll and Ghost In The Shell, they are far less sophisticated than Akira. Since Akira, the only other "anime" films I'd put on the same tier as Akira, or close to it, would be the films by Miyazaki, which are remarkable for their animation. But I should say, I consider all these films, the same way I consider anything Pixar does, a film. I don't believe that there is a difference between such things as a movie or an animated movie. Movies are movies to me.
As for anime shows, the only one I would put as high as any film variations is the miniseries Berserk. Berserk is just exceptional, even if the animation has more in common with standard anime (in terms of technical stuff) and lacks the streamlined, fully dynamic touch seen in Akira or Miyazaki movies. The story and characters are what this one does well, and characters like Guts, Casca, Griffith, and the others are as memorable to me, or more so, than the ones in Akira.
Pretty much all the ones you guys mentioned in this thread besides those, I've seen. Not really for me. I've also seen quite a bit of Inuyasha, since my girlfriend has an obsession with it. I understand how they can be fun to watch, but they aren't very compelling.