If you wish to travel by air, the nearest airport is Jinju's Sacheon Airport. From the airport, catch a limousine bus to Tongyeong.
There are no direct trains to Tongyeong. The closest train stations are at Jinju, Haman, or Masan, and you'll have to take a bus from there.
The most convenient way to get to Tongyeong is by bus. Buses travel regularly from both Seoul's Nambu Terminal at Seocho (11 times daily) and the express bus terminal at Gangnam (14 times daily). The ride takes about 4 1/2 hours. There are 8 express buses and 14 regular buses a day from Daejeon and about 18 buses from Daegu's Seobu Terminal. Buses from Busan's Seobu Terminal run daily from 6:30am to 10:30pm about every 20 minutes. From Masan's Nambu Terminal, there are daily buses from 5:10am to 9:15pm every 10 minutes. Buses from Jinju run about 63 times a day from 6am to 9pm.
Driving from Seoul, take the Gyeongbu Expressway to the Tongyeong-Daejeon Expressway, heading toward Tongyeong to the Tongyeong IC. From anywhere in the Gyeongsang-do area, take the Jungbu Expressway to the Namhae Expressway to the Tongyeong-Daejeon Expressway, heading toward Tongyeong. From Jeolla-do, take the Honam Expressway to the Namhae Expressway to the Tongyeong-Daejeon Expressway, heading toward Tongyeong.
There are local buses available to get you where you want to go, but Tongyeong's major attractions are all within walking distance of each other.
You can catch a ferry to some of Tongyeong's outlying islands or an excursion tour around the area. The best thing to do is to get one early in the morning so that you'll have plenty of time to explore. Island destinations from Tongyeong's Ferry Terminal (tel. 055/644-0364) include Dumi-do, Maemul-do, Mireuk-do, Bijin-do, Yeondae-do, Yeonhwa-do, Yokji-do, Chu-do, and Chubong-do. Tour boats usually leave from the Excursion Boat Terminal in Donam-dong, while long-distance boats dock at Ganguan Port.
There is a tourist information office (tel. 055/650-4584) outside of the bus terminal and it's open Tuesday to Sunday from 9am to 8pm (closed on national holidays and the day after). The staff's English isn't great, but they do have maps in English. There is also a tourist information booth at the excursion boat terminal at Donam-dong (tel. 055/644-7200) and at the Hakseom Resort (tel. 055/640-5245). The city's official site is at http://eng.tongyeong.go.kr.