There are regular ferries between Mokpo and Jeju-do, and also to some smaller islands in the southwest. Ferries to Jeju cost between 25,000 and 49,000 won, depending on how fast you want to get there. Cheaper ferries take about 5 hours, and faster ones can make it in about 3.
Trains run regularly to Mokpo, which is the last station on the rails in the southwest. The KTX line splits in Daejeon*, and only the Honam line continues west. Thus, passengers heading to Mokpo must travel through Seo(West-)Daejeon station. From there, the trip takes just over two hours on the KTX and costs 25,000 won.
The intercity bus terminal is a 10-minute (5,000-won) cab ride from the ferry terminal and train station. City bus 1 also runs this route, and costs 1,100 won.
Please note that trains out of Seoul come into the Daejeon main station. Travel from Daejeon in the direction of Mokpo goes through the Seodaejeon Station. Attempting to book a direct train from Daejeon to Mokpo on the Korail Website will produce an error.
Yudalsan Park - Yudalsan is well worth a visit for the commanding view from its summit of both the bustling city and the many scenic islands. On your walk up, be sure to visit the noon gun, installed during the Japanese occupation, and two beautiful ""Buddhist rock carvings"".
Sculpture Garden - On the east side of the ring road surrounding Yudalsan is an impressive sculpture garden showing about 50 large pieces from around the world.
Gatbawi Literature Hall- featuring information about famous writers from Mokpo. No English available. 2,000 W.
Moe's Bar and Grille, Rose Street (Upstairs from Eleven AM Cafe.). 8PM-2AM. Nice western bar once run by a guy from California. Offers a few kinds of beers and cocktails. Free popcorn.2,500-10,000 won. edit
Kumho Beach Hotel : 40,000 won. Clean and modern rooms. Cable, Internet and PC in the rooms. Five minutes from the train station and not far from the ferries. 061 2425700. Kumho is a very tall building which can be seen from a distance. If you go to the surrounding area, there are many smaller lovelier motels.
Shinan Beach Hotel : 100,000 won. Regarded as Mokpo's top hotel. Rooms are mainly "ondol" floor-heated, or twin. 50% discount during off-peak season. 061 243 3399 
Hadang Boseok Sauna Jjimjilbang, 1000-4 Sang-dong, Mokpo-si (As you walk north from Lotte Cinema, you can see the sign on the left, behind other buildings, from the main street.), ☎ 061-281-8102. >10,000 W/night. This jjimjilbang is located next to a small park surrounded by buildings near a port.The jjimjilbang clothing is very comfortable, but the common area itself is ok. The floor (in July) is hot enough to make you sweat. There are no floor mats, and the head rests are made of wood and uncomfortable. Try sleeping in an area up some stairs with a low ceiling, which has open windows and a cool floor, which is very comfortable. The restaurant is decent. There is a room for sleeping which has outlets for charging phones, but the air is stuffy. There is another jjimjilbang nearby, near Lotte Cinema.edit
To continue on to the islands to the south by bus, visitors must depart from the intercity bus terminal. A ticket on a coach bus to the island of Jindo costs 6,000 won and the trip takes just over an hour. This bus runs every 30-45 minutes from 6:15 - 20:30
You may also want to consider a trip from the city's passenger ferry terminal, from which boats run to a dozen nearby islands, many of which are part of a marine national park. Heungdae (about 40,000 won round trip) and Hongdo (about 75,000 won round trip) are particularly beautiful.