Adisucipto International Airport, Yogyakarta
The airport was severely damaged in the 27 May 2006 earthquake, and was only completely rebuilt in 2008. Yogyakarta is the only airport in Indonesia that has a train station. If you're heading first to Solo, then the train is a fast and cheap option to get there.
As with all Indonesian airports, those traveling into or out of Yogyakarta are strongly encouraged to lock their luggage. Pilferage from checked bags has been a problem at most Indonesian airports. Also note that the ICAO liquids, aerosols and gels rule applies to all carry-on bags for all departures from this airport.
For ideas about what to see and do around the city, see the Asia for Visitors guide to Yogyakarta.