I don't know how many days you plan to spend on Jeju, but I'd recommend at least 3 if you're going to be doing any outdoor activities.
Hallasan (the highest mountain in South Korea) is a nice hike in the centre of the island and is worth doing if the weather cooperates. There are a few different trails you can take, but I believe a few of them are currently closed to allow them to recover. I took the Seongpanak Trail which starts on the east side of the mountain. I think it took me about 6 or 7 hours total up and down, and I'm a steady but not a fast hiker. You can catch a bus from Jeju City Bus Terminal that drops you off right at the trailhead on highway 1131.
I'd also suggest going to Seongsan on the east coast which has a huge caldera(?) that is quite interesting and you can walk right up to the top to look inside. From Seongsan you can also take a short ferry (maybe 15 minutes, usually about once an hour although that may be a lot less at that time of year) to Udo Island. Upon arrival on Udo, there's a place that rents mountain bikes and scooters which you can use to explore the island. Definitely worth checking out.
Finally I'd suggest walking a section of the Jeju Olle Trail. This is a trail that almost encircles the island and is broken into stages of different lengths that usually take about 5 to 9 hours. I've walked most of two sections. The first was in the southwest of the island, starting from Sanbanggul and the Yongmeori Coast. Yongmeori is a beautiful section of coast and from here I walked counterclockwise towards Seogwipo. The other section was in the southeast and I started at Pyoseon Beach and walked clockwise, also towards Seogwipo. The nice thing with these trails is that other than the odd tour group, they're relatively deserted (which is a nice treat if you've ever hiked in a Korean NP on a weekend) and you get a wide variety of scenery.
Here's a link to the official Jeju Olle site - it's in Korean but you can see a map of the different sections. When you arrive at the airport in Jeju City you can pick up an English map just inside the arrivals area.
[https://www.jejuolle.org/main/main.jsp]
Here's another link from the Korean Tourism Office:
[http://visitkorea.or.kr/enu/SI/SI_EN_3_4_12_13.jsp]
Jason