Even though it has many names the Mokpo Bus Terminal is the only bus station in Mokpo that you can use for your escape. Sometimes it goes by Distance Terminal, Express Terminal, or Cross-Country Terminal, but everybody (taxis included) recognize it as the Bus Terminal. Here's the contact information:
전라남도 목포시 상동 220As usual, trying to find English information about the bus schedule is frustrating, leaving us no option but to try and interpret what exists in Korean. Right away we have two starting points: EasyTicket and The Kumho Asiana Group.
(Jeollanamdo, Mokpo, Sang-dong 220)
☏ Tel : 061-276-0220 or ☏ Tel : 061-276-0221
With no surprises EasyTicket turns out to be a ticketing agent for Kumho's bus division and Kumho's bus division (also known as Kumho Express) turns out to be the bus company that seems to run all the buses out of Mokpo. But it's not quite that simple. Kumho operates the www.kumhoexpress.co.kr (in English) website, but Kumho is also listed on the Express Bus Lines Association's www.kobus.co.kr (in English) as well; this page on KumhoExpress lists EasyTicket and Kobus as handling the tickets for different routes. So what site you use depends on where you're going. Now, within this Kumho bus monopoly in Mokpo there are two kinds of buses useful for escape: direct and express buses. The main difference here is based on how far you're going; Mokponians use a direct bus to escape to the neighboring towns and an express bus to escape to the neighboring Seoul. So, in the end, it seems that direct bus users can use KumhoExpress while express bus users should use Kobus. But either way you'll need a Korean credit card to make the purchase and that's something that most foreigners don't have (i.e. cannot legally obtain) leaving the only option of a cash purchase at the ticket counter.
That doesn't mean that the sites themselves aren't useful. The English version of KumhoExpress does provide a timetable but it is outdated and hasn't been touched since 2003; the more recent Korean version does provide a direct bus trip planner that lists the options for short trips like the popular Mokpo to Gwanju route:
English trip planner.