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).
When checking-in for flights from Jakarta, you will need to pay the departure tax to the ticket agent before receiving your boarding pass. The departure tax is 50,000 Rupiah (6.00 USD) for domestic flights and 150,000 (18.00 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 50,000 Rupiah (6.00 USD) for domestic flights and 150,000 (18.00 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.