Gwangju has over 80 city bus routes, and most run past the bus terminal that has bus stops on all sides. Bus 17 (W900, 20 minutes, every 15 minutes) runs between the bus terminal and Gwangju train station.
The subway (W800, every 10 minutes) is being extended west to the airport. Buy a disc ticket and at the subway barrier hold the disc ticket over the yellow pad. When you reach your destination put the disc ticket into the slot at the exit barrier.