The basic requirements are simple ( http://www.nasajobs.nasa.gov/astronauts/content/broch00.htm#bqr ). What actually sets some people apart from others, I don't really know. If/when they decide to go to Mars, or even back to the Moon, I would think that a degree in something like Planetary Science might be useful to get selected. Of course, physics, and other similar degrees too.
With the "3 years of increasing responsibility" requirement, you should probably try to make it as close as possible to the space program. Or, they seem to like sending teachers up, maybe you could do that too ;)
These are all just guesses on my part. I plan on applying too. Perhaps we'll both be stepping onto Mars one day :)