For those using public transport, bus is the most convenient way to get to Donghae for most people. The bus terminal is a 20 minute walk north of the downtown area (5 minute cab ride.) There are frequent buses to the Express Bus Terminal in Seoul (takes a little over 3 hours), and less frequent buses to Dong Seoul (East Seoul.) There are also buses leaving every 10 minutes for Gangneung (35-40 minutes) as well as buses to most other cities in Gangwon Province. There are 3 overnight buses to Busan.
The Yeongdong line goes through Donghae, stopping at Donghae Station (south of downtown) and Mukho Station (north of downtown.) There aren't a lot of trains, however, and those there are don't go to too many places that are appealing to tourists, making the bus more convenient for most tourists.
Donghae serves as the main access point to Ulleungdo, with regular ferry service throughout the year (usually once or twice a day in each direction, more frequent in the summer.) Ferries leave from Mukho Port, a 5 minute drive of downtown.
Mureung Valley (무릉계곡)
This is a fantastic valley with several hikes of varying difficulties. You can hike to waterfalls, temples or up mountains. Admission to the valley is 1,500 won for adults. The valley can be reached by taking any of the city buses numbered 12-something (e.g. 12-4.) The bus fare is 1,000 won.