i don't want to be rude, but very nearly every single foriegner in South Korea has travelled to Jeju. And so has every single South Korean. You question is so vague it is hard to answer. With the variety of experiences available on Koreabridge regarding Jeju you would do well to describe what you like to do and when you intend to go.
I am not a foodie but if you ask, people here will tell you where to get the best raw fish or black pork or citrus fruit, etc.
I love hiking and if you have a day to do it, I think you should climb Hallasan.
In May, the weather will be excellent for human powered travel everywhere - ride a bike around the island on the excellent trails, hike or look for sea kayaking.
I might swim in Jeju in May, but I am eccentric that way. I don't think many people choose to swim in May there. If you can, I think the beaches on the South side are a little better but they are all good.
There are roughly 500 little tourist attractions on the island - I mean niche museums, Ripley's Beleive it or not, Smurfworld, Ghostworld, Horseworld, Darkride World (I made up the names of the last four but I bet similar attractions exist). Ah, more objectively (the formatting looks terrible in the typing window. Follow the link for the names and links to many little museums and attractions and more):
Museums and Art Galleries