(This post is in response to Annick's questions, thank you for the idea.)
When you arrive in Incheon airport it's super easy to just hop on the subway with your bike, however that is not the case in Busan. Gimhae airport is located just West of Busan city and it is connected to Busan by a light railway train. It is an automatic elevated train and no bicycles are allowed on it.
So how to get from Gimhae airport into Busan? Or how to get from Gimhae airport to the Nakdong river path leading to Seoul? Or how to get from Busan to Jeju by plane? Well of course you can take a taxi, if the driver agrees and if you take your wheels off. Some airport buses would probably let you store your bicycle under the bus in the luggage department.
But the best option yet is to simply ride your bike.
Few people are aware that the airport is 1,5km away from the Nakdong river bike path. So with a little GPS and Google or Naver maps it should be super easy to navigate yourself to the bike path.
Once you are on the path you can cycle down the West bank all the way to the first certification center, it's about 10km. If you happen to be doing that in the last week of March or the first week of April you will cycle through a 10km tunnel of Cherry blossom heaven on a dedicated bicycle path.
Near the end of the lane of trees the path will turn to cross the river and you will soon see a sign for the certification center. After the certification center you can cross the rest of the river and start the route on the East bank of the Nakdong river and head to Seoul. Or you can cross the bridge and go straight for 1 km to reach Hadan subway station. From Hadan you can take the subway to anywhere in Busan.
If you cycled from Seoul, ended in Busan and want to take a plane to Jeju, you can cycle from the certification center to the airport.
Any information about cycling in South Korea.