Azumanga Daioh is quite funny, once you get past that it really is just about a few high school girls. No saving the world, no supernatural powers. It's kind of got a random humour element to it. Kind of like Excel Saga, but less extreme.
Also in the realm of the random is FLCL aka FuriKuri aka Fooly Cooly. Shorter than most anime at 6 episodes, it still tells the story quite well. And the music is outstanding.
Can't say enough about Cowboy Bebop. The story and music are superb on their own, but the way they're meshed together just brings it to a whole new level.
Also worth checking out, purely because the music is done by Yoko Kanno (the same person from Bebop) is Wolf's Rain.
Mahou Sensei Negima is also pretty funny, although it's strung together by supernatural powers etc. Also about (mainly) high school girls and their 10 year old teacher. I haven't seen the remake they have/are recently doing/done, but it seems to cut out most of the fanservice shots, and a bit of the humour.
Kinda surprised no one's mentioned Last Exile. A steampunk aviation centered anime, with a great story.
Also Hunter X Hunter, which is a feel-good anime about a 12 year old boy and his travels through his world.
Really surprised no one's mentioned anything Ghost in the Shell. While the original movie is worth mentioning just for the thought provoking situations it presents, the Stand Alone Complex series (both seasons) and the Innocence movie are some of the most visually pleasing and best storytelling works of art I've seen.
Ergo Proxy is a new one in the same sort of vein as Ghost in the Shell.
Samurai Champloo is kind of a new take on that Meiji Era wandering traveler anime-cliche. Done by the same director as Cowboy Bebop.
I've downloaded Rozen Maiden, but haven't watched it yet. Have heard a lot of good things about it.
If you're looking for better summaries, I'd recommend animenewsnetwork.com, or animenfo.com. anidb.net also has listings of some of the fansub groups that have released that anime.
As far as movies go, I'd definitely recommend Akira and Princess Mononoke. Akira is set in the near-future, while Princess Mononoke is set in more of a earlier setting. There's guns, but no cars, etc.
And to Excal, stop watching anime in english. go to anidb.net, find some decent subtitled releases of Ghost in the Shell, Cowboy Bebop, or Kenshin Tsukiohen (the first OVA, also might be referred to as Samurai X OVA, *not the motion picture*).
