You could visiting the most southern island called Mara island in South Korea from Jeju island.
You may enjoy fishing during boat trip to Mara island.
Please try to eat Jajangmyeon which is a noodles with special sauce made from seafood.
톳해물짜장짬뽕
43 Mara-ro, Daejeong-eup, Seogwipo, Jeju-do
064-794-0489
You also enjoy sailing at Southern part of Jeju island.
요트투어(주)
2950−5 Saekdal-dong, Seogwipo, Jeju-do
1544-2988
There are many good visiting spots in Jeju island. Here is a list of my recommendation.
https://www.google.com/maps/placelists/list/3iW31mqwlbeqVRHiK7GHhrZ44rozaA?hl=ja