Vision of Escaflowne is an easy show to get into, and a classic at that. Along with Trigun, they were the two shows that really got me interested in anime.
As for other suggestions:
Baccano
If Tarantino were to make an anime, I would think it'd be something like this. There are multiple non-linear stories, sudden bursts of intense violence, and quite a bit of humor. The characters range from immortal alchemists to gangsters. Also, the Japanese voice actor for Vash from Trigun has a role in this show as one of the two characters that provide comic relief.
Great Teacher Onizuka
This show sounds kind of stupid (it's about an ex-gang member and general felon who desires to be the greatest teacher in Japan,) but it's really rather funny and the show supports an enormous cast of characters that all get fleshed out over the course of the show. Basically the main character, Onizuka, interacts with the students in his class and tries to make school fun for them in a way that teachers neglected to for him.
Time of Eve
This is a very short show (6 15-minute episodes) but it's exceptionally well done. From an animation studio run by a handful of people, Time of Eve takes place in the future when robots are part of everyday life. They follow Asimov's 3 Rules of Robotics, and the entire show is about a cafe where both robots and humans keep their identity secret. (Every other place in the world robots are required to have a halo above their head to show they aren't human.)
Planetes
In the year 2070, following a group of trash collectors in space, this show tries to accurately portray how it is that humans would expand into space, and the psychological effect that life in space has on an individual.
Now, my favorite anime happens to be RahXephon, but as far as introductory shows go, that's not a very good choice. If you've heard of Neon Genesis Evangelion (which I haven't seen,) they're often compared, but if you are up for a real mind-bender, I guess you could try RahXephon. The plot is more or less about a young man named Ayato who finds out that the world (Tokyo) was a different dimension separated from the rest of the world. He has to deal with this, along with his ability to pilot the robot Xephon which has the power to tune the world and make it anew. From there the plot is carried along by character development, and that's what makes the show so good. Although there is token robot fighting, it only serves as an environment for Ayato to grow as a character, and not to serve as excitement.
