Swim
Deokpo Beach
A very small sand beach just out of Okpo.
Geoje has a tonne of swimming beaches around its coast.
Gujora and Wahyeon Beaches
A pair of large sand beaches overlooking some pine-covered islets. Gujora town itself is rather interesting too as the locals have undertaken a project to paint art on every wall in town yielding rather pretty results.
Hakdong Black Pearl Pebble Beach
Just a rocky pebble beach really.
Geoje Art Center
A theatre on Geoje, check local schedules. Hotel attached.
hike
Being Korea, Geoje is covered with hiking trails.
If just looking for a casual stroll, the 4km long trail from Okpo to Deokpo is very pleasant and consists of trails along the coast through pine forest and segments along red boardwalk where you an see older ladies foraging for shellfish for the local restaurants. Passes by the Great Victory Park and Deokpo at the end hosts a couple of hoe sashimi restaurants and a coffee shop for refreshments. One of the restaurants doesn't welcome foreigners, but the one two doors down does and is actually quitely literally signposted that way. Starting point is at the north end of Okpo port, look for the ruined themepark on the cliffs.