Jeju Island has many natural spots including Halla Mountain.
Easy to explore by public transportation and bike.
Whenever I visited Jeju, I used to rent a car because the public transportation was not good and taxi driver tried not to put the meter on properly (it doesn’t happen thesedays). However, in 2017, Jeju authorities changed the bus system totally favorable to tourists, therefore you can go everywhere with T money card easily. But still rent a car is best option if you want to visit several place in short period with kids.
Jeju Island has some sad stories in Korean history in the past. It used to be a place of exile in Joseon Dynasty (14 C to 19 C), the residents in Jeju could not leave the island at that moment, they resisted strongly against the separation of two Koreas in 1948, then the holocaust was set up by the army, which results in lots of victims, etc.
Nature, History, and Food.
4.3 Peace Museum is a must to visit if you are interested in local history. This museum is very well organized. After the 4.3 massacre, many Jeju people escaped from Jeju to Japan in order to save their life. Many descendants live in Osaka area now. https://goo.gl/maps/RTsEMNu6wGu
Jeolmul forest is close by 4.3 Peace Museum. Less than 30 Min walk distance from the museum. Mostly, local people come to take a walk, hike or take a rest. What I liked inside, there is wood house, where you can take a nap under the sunshine.https://goo.gl/maps/bY5bDcy7HVo
Kim Man Deok, a lady merchandiser who saved Jeju people, starving from the notorious drought around 1792-5. The government didn’t do anything to save the residents at that moment, so she released her property to save local people. Afterwards, King wanted to thank her, asked her if she has any wish she wanted to do. She wanted to get out of Island once and visit the Peninsular. She visited Geumgang Mountain (located in North Korea now) and met the King. Her grave is in Mochung temple. I recommend visiting her memorial with Kids. Good facility for kids. Memorial is close to Dongmun Market in jeju city. https://goo.gl/maps/Fka4VC8gzzQ2
You can try some food at local traditional market, such as Dongmun Market. Seafood is good to buy here. Very fresh but a little bit pricy. You can order to deliver to Seoul or any other cities in Peninsular from here.
Artist Street
Lee Joongseop is one of the most famous artists in Asia. He was falling in love with a Japanese lady under the sad period of Korea. They had to separate because of different nationality. Before that, they lived in Jeju together with 2 kids. Someone lend a small room (about 3.3 cubic meters), 4 family members lived there happily even if they were poor. He passed away in Jeju missing his wife and kids who had to leave for Japan after the WWII. War made people’s life in misery. Mostly he drew sketched his lovely family when they were together. Behind the house he lived, there is a small gallery where you can see his arts. He passed away in his early 40’s. https://goo.gl/maps/Zy1x8T6NbQ22