Before doing anything else, clean the battery posts to help ensure a good connection to the starter.
Check to make sure your battery is holding a good charge before possibly wasting money on a replacement starter. The voltage should be around 12.4.
Refurbished starter
When removing the stater, first disconnect the negative battery cable, then disconnect the positive battery cable and the other little wire from the starter. I also disconnected the air hose and pushed it out of the way. The starter is held in by two large bolts. Using a breaker bar helped us get them out without stripping them.
Here's the old starter. I took it apart so I could see how it works, but I ended up pretty much destroying it by damaging the wire connections while trying to put the solenoid back into the spring-loaded magnets. They still gave me the core charge back though!
This refurbished Mitsuba starter did not work properly, go figure. I had to take it back out, drive the 15 minutes back into town and exchange it, and then put the new one in. Thanks a lot Mexico QCers.