ROK has seven international airports: Busan (Gimhae Airport), Cheongju, Daegu, Jeju, Muan and Seoul (Gimpo Airport and Incheon International Airport).
Incheon International Airport, about an hour west of Seoul, is the country’s largest airport, with good connections throughout the world.
There are direct inter-city buses to many locations throughout ROK just outside the international arrival hall and you can buy tickets at the airport. The airport has a direct rail service to Seoul station by the express and all stop AREX (Airport Railroad Express) trains. The express service takes 43 minutes to reach Seoul Station and offers a more mainline train journey.
The all stop train takes a slightly longer 56 minutes and offers a more subway like journey.
You can reach the airport by the AREX train, an Airport Limousine bus, or by line 5 or line 9 of the Seoul Subway.
Seoul Gimpo airport is more centrally located next to Seoul than Incheon, and offers primarily domestic flights to most South Korean cities. Busan’s Gimhae airport (PUS) has international flights to Cambodia, China, Guam, Japan, Hong Kong, Laos, Malaysia, Philippines, Russia, Taiwan, Thailand and Vietnam.
If arriving/departing from Incheon Airport on Korean Air it is possible to book connecting flights to/from Busan though the airline, note that as immigration is completed in Busan, it is not possible to leave the Incheon Airport and baggage is checked right through to Busan. Air Busan and Korean Air offer flights from Seoul Gimpo to Busan which can often be cheaper than the KTX.