Tsushima (対馬 Tsushima) is an island in the Tsushima Strait (the eastern channel of the Korea Strait), about 50 km from Korea and 130 km north of Fukuoka. The rural island has a population of around 34,000 and approximately 90% is covered in mountainous forest. The city's English webpage  offers a significant amount of historical and tourism-related information (including restaurant maps and recommended travel courses).
ANA  flies to Tsushima from Fukuoka and Nagasaki. Prices change depending on type of ticket, but generally run around 13000-15000 yen.
Fukuoka ⇔ Tsushima
Runs 4 round trips per day (30 minutes)
Nagasaki ⇔ Tsushima
Runs 4~5 round trips per day (35 minutes)
The Ocean Flower 2, the ferry from Busan, Korea operated by Dae-a Express Shipping, operator of the ferry linking to Tsushima Island, Japan. carries passengers only between Busan and Hitakatsu in 1 hour 40 minutes and between Busan and Izuhara in 2 hours 40 minutes. 
Rental car is the best way to see the island, as the bus system is fairly limited and does not directly visit most tourist sites. However, renting a car may require some Japanese ability and costs around 5000 yen a day.
For those wishing to use the bus, there is a one day bus pass available to foreign tourists for 1000 yen.
There are also a number of taxi companies and some bicycle rentals available. The city's page offers more extensive information about transportation in Tsushima.
Famous things in Tsushima include the Tsushima Yamaneko "Tsushima leopard cat" and "Arirang Festival." Although amalgamated into one municipality in 2004, Tsushima-city was once divided into six towns. The towns still have some autonomy and are often used to tell where a location is on the island. The city's page has an extensive list of sightseeing spots grouped by each town.
Arirang Festival (アリラン祭り). Arirang is a Korean friendship festival, held in Izuhara on the first weekend of August. Highlights include a parade showcasing traditional 17th-century Korean and Japanese costumes, stalls with usual Japanese matsuri snacks, and a large fireworks display on Sunday night. This festival has been suspended for 2013 and possibly longer due to the incident of five South Korean men stealing Buddhist relics which had been on the island for more than 3 centuries. edit
Tsushima Wildlife Conservation Center (対馬野生生物保護センター), . The center exists primarily for the conservation of the Tsushima yamaneko (Tsushima leopard cat), but also provides information on the ecology of Tsushima in general. One yamaneko is usually on display, but can sometimes be hard to spot in the enclosure. Admission is free, and the center is closed on Mondays.edit
Ayumodoshi Nature Park (鮎もどし自然公園). Ｌocated about thirty minutes’ drive south of Izuhara, Ayumodoshi is a large park area set among the nearby hills and streams. One stream in particular cuts across a bed of rock, creating small rapids. The park’s name refers to sweetfish (Ayu) having to turn around because of the rapids. The stream has carved pools in the rock, many of which are suitable for swimming. A suspension bridge spans the stream at one the park’s entrances, and there are many trails around the park. The trail for climbing Mt. Tatera also starts at Ayumodoshi. Camping facilities are available. edit
Tsutsuzaki - Osakiyama Park (豆酘崎・尾崎山公園). The southern tip of Tsushima affords a beautiful view of the sea. There’s a lighthouse just off the cliffs from the vista, with rocks jutting out of the water along the way. edit
Beaches. Tsushima has a number of beaches scattered throughout the island. Miuda beach （三宇田浜） on the far northern end of the island was chosen as one of the 100 most beautiful beaches in Japan in 1996. Miuda has camping facilities available. A bit further south, Mogi beach (茂木浜) also has pristine white sand and emerald green waters. About five minutes from the airport, Green Park has a pair of beaches that attract a significant number of visitors and are good for snorkeling. South of Izuhara is a rocky beach called Oura (尾浦） that has camping facilities and barbecue pits. All of the above-mentioned beaches have restrooms and showers available for use in the summer swimming season.edit
Tsushima Family Park (対馬ファミリーパーク). Located in Mine, in the center of Tsushima, the Family Park is just off Route 382 (the island’s main road). It features roofed basketball courts, a go-kart track, and a sprawling playground. There’s a tall hill at the center of the park, which provides a great view of the surrounding inlet. Part of the hill is carpeted with artificial turf, and plastic toboggans are available for sliding down the hill. There is also a large slide that ends at the playground. edit
Hiking, Mountain Biking & Fishing are the main activities whilst on the island.
Hiking. Tsushima has a number of mountains that can be hiked in a few hours or less. Shiratake (白嶽, 519m) is the most famous peak on Tsushima. The aptly named Joyama (城山), meaning "castle mountain," contains ruins of a 7th century mountain fortress. Mt. Tatera (竜良山, 558.5m) is famous for a primeval forest near its base that has trees over 1000 years old. Other popular hikes include the 558.2-meter high Mt. Ariake (有明山) and Mitake (御岳) at 479 meters.edit
Biking. Cycling is popular on the island, with many tourists choosing to enter the island through the northern port, Hitakatsu, and bike their way down to and depart from the southern port, Izuhara, and vice versa. There is also suitable terrain for mountain biking, but the trails are unofficial and require a bit of local knowledge.edit