The big five for first-timers
Seoul is where almost every trip begins: palaces, hanok lanes, markets and a nightlife that runs until dawn. Give it at least four days.
Busan is Korea's second city and its seaside soul, with city beaches, a huge fish market and the painted hillside of Gamcheon. The KTX gets you there from Seoul in under three hours.
Gyeongju was the capital of the Silla kingdom for nearly a thousand years. Grassy royal tombs, Bulguksa Temple and the Seokguram Grotto make it Korea's open air museum, and it sits close to Busan.
Jeonju is the home of bibimbap and has the country's largest hanok village. It is an easy day or overnight trip from Seoul, and the best place to taste traditional Korean food culture.
Jeju Island is a volcanic island an hour's flight south, with lava tubes, crater hikes, black rock coasts and the haenyeo women divers.
How to combine them
A classic ten day loop is Seoul, Jeonju, Gyeongju and Busan by train, then a flight from Busan to Jeju and home via Seoul. With a week, choose Seoul plus either Busan and Gyeongju or Jeju. Use the travel tools to search flights and stays, and the places to visit list for food and sights in each city.
If you would like help shaping a route, the free trip planner is the place to start.