Haegeumgang
Designated "Korean National Scenic Site #2, this is a must-see location. Essentially just a locally famed island consisting of tall cliffs. However, the ferry cruise around the island will actually thread its way through the extremely thin cliffs leading into the island into a sea cave before backing out and departing through different route. Spectacular. Round trip three hour cruises including both Haegeumgang and a 1.5 hour stopover on Oedo from Jangseungpo will set you back 19,000 won for an adult and 12,000 won for a child. A 2000 won discount applies in off-peak season. Note that actual admittance on to Oedo incurs an extra fee and inability to pay will leave you at the boat twiddling your thumbs.
Oedo Island
Typically included in cruises to Haegumgang, this is Korea's first privately owned island and host to the Oedo Botanica. It's somewhat of a cross between a botanical garden and one of East Asia's infamous "country-themed themeparks" and designed to make you think you're somewhere in the Mediterranean. Beatifully landscaped and covered with greco-roman statues. Actually a bit prettier than the brochure describes as the gardens follow a tight valley surrounded by sheer cliffs. Take a detour to the bathroom as the off-the-path views from outside a couple of them are among the best on the island. Conceived by a couple that crossed the 38th parallel after the war quite literally penniless. Worth the trip and has a restaurant and coffee shop at the highest point that mostly only serves tea Udon 4000 won, coffee 4000 won, tea 2500 won.
Great Victory Park
A war memorial for the 1592-1593 naval battle against the Japanese. Next door lies a shrine to Admiral Yi who lead the Korean forces to victory plus a small museum explaining the details of the war and Korean naval strategies. English explanations exist, although when browsing try to keep count of how often you read extremely biased language answer: a lot.
Windy Hill and Shinseondae
A picturesque cliffside hill with a windmill on one side and a bunch of purple-green cliffs you can climb down on to on the other side. There is also a Theme Museum here. At the time of writing the themes are "Korean modern life" and "European decorative art" aka mannequins of pirates. The intercity buses to Tongyeong and Masan depart from here.