(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.
In March I at last fulfilled my goal to not only see Jeju in spring time, but also to cycle around it. A co-worker joined me in my quest to cycle the +/- 250 km around the coast.
We decided we could not be bothered by getting our bikes boxed for flying, so we booked a sleeping bunk on the 12 hour ferry from Busan Ferry Terminal to Jeju.
I booked the tickets online and just picked them up at the port, we put our bikes in the hold of the ship, went on board and made ourselves cozy in our room. The next morning we got off the boat, got on our bikes and started cycling.
Our adventure started at 6:40 am with lots of black rocky coastline, cute artsy cafe's and guesthouses, some white and black sand beaches, 3 or more waterfalls, an irritating amount of wind (on the last day), just enough Chinese tourists and Korean school groups, modern wind-mills dotting the coast, endless volcanic stone walls surrounding the similar black stone houses, drying squid flapping in the wind, women-divers' scooters and carts parked along the road galore, yellow rapeseed flowers lining every road and tourist site, with some tulips, magnolias and other blossoms thrown in between, wide safe cycling paths as far as the eye can see, clean well stocked bathrooms around every corner right when you need them and many kind smiles and words to accompany us on our way.
Oh and lets not forget sore cycling bums, a wet rainy cycle and numerous CU or GS25 pit-stops, delicious grilled fish and black skinned local bbq pork, then throw in red and white lighthouses, some horses, too many scary chained dogs, a boat ride or two and mystical dramatic rock formations showing off the ancient power of the earth's crust.
Thank you Jeju, Korea and my cycling partner for such a wonderful adventure!
Some tips and info for cycling Jeju:
Any information about cycling in South Korea.