I did a bike trip around Jeju. It is quite doable, as it is small. For those who are used to biking it is a cinch, as people (drivers) are 99% nice (other than the one who honked at me for biking on the road, which is part of the bike path, and the bus driver who almost hit me). 3-4 days.

Everywhere along Jeju you see the coast (I guessed I biked the coastal route) and there are always cafes to get a great view (or you can go to 7-11 and save a few bucks if you want) and there are many hot springs to relax in at night. English isn't spoken much, Chinese is more useful (lots of Chinese working there).
They have good seafood, and actually I had a great burger somewhere on the east coast.