Apart from Seoul there are also other cities that are very popular for tourism.
Nami Island 남이섬 is a filming location from a famous Korean drama called winter sonata. Its basically like a huge park where couples and families go take a stroll, ride bikes and take pictures with the famous tree lines. Aprox. 2 hours from Seoul, fee is 10,000 and you have to take a ferry from Gapyeong to reach the island.
Jeju Island 제주도 is one of the biggest islands of Korea, a volcanic island with Hallasan as one of the highest points and a popular destination for honey moons. You can also visit the Sex Museum and see other weird-looking statues. Jeju is also famous for its oranges/tangerines and Jeju Black Pork (오겹살 and 항정살 cuts).
Busan 부산 is one of the biggest ports in Korea and famous for the beaches. Heundae beach, the Buryeong market, Gamcheon cultural village, etc are some of other attractions of the city.
Yeosu 여수 has recently become a very famous destination. The Expo 2012 regarding the Living Ocean and Coast, Preservation and Sustainable Development, etc received a lot of attention.
At DMZ you can visit the border with North Korea and see the fake folk village on the other side of the line.
Snoraksan 설악산 is the third highest mountains in Korea. Since Korean people love hiking this is also a very popular destination.
The Boryeong Mud Festival at Daecheon beach held every year during July is one of the biggest festivals visited by tourists.
During my last months in Korea I visited 전주 Jeonju which is considered the foodie city. We used a discount program from the train company that offered cheaper prices for students. When we went there we saw a lot of young people with their selfie sticks taking pictures with food. The architecture is also very nice and traditional called 한옥마을 hanok maul. We stayed at a traditional Korean home Saekdong Jeogori 색동저고리 for the night. It was very cozy and convenient.
As you can see there are many other destinations apart from Seoul. Transportation in Korea is very developed so you can take either a bus or a train to reach your destination. KTX is my favorite which is the fastest train but also quite expensive. There are apps also available for booking online or just to check the schedules. Safe travels!