On arriving at the airport, be sure to have your ticket and passport handy. You'll need to show them to security before being allowed into the check-in area of the terminal. You'll also need to have your baggage scanned. If departing on a domestic flight, you also need to have your checked bags strapped at the security stations next to the baggage scanners (free service).
After checking-in for flights from Yogyakarta, you will need to pay the departure tax to the ticket agent before you can get into the departure lounge. The departure tax is 25,000 Rupiah (208,333,333.33 USD) for domestic flights and 100,000 (833,333,333.33 USD) for international flights. After checking in, international passengers will need to pass through immigration. There is a fiscal check point before you get to immigration, but this is for Indonesian residents only. You may be asked to show your passport, but usually there is no need to stop at the counter.
The departure tax is 25,000 Rupiah (208,333,333.33 USD) for domestic flights and 100,000 (833,333,333.33 USD) for international flights. Note, however, that as of 4 October 2012, the domestic departure tax for Garuda Indonesia customers has been included in the ticket price and does note need to be paid at the airport.