Trains
South Korea's rail network is excellent and getting around by train is the most comfortable way to cover longer distances. The KTX high-speed train connects Seoul to Busan in about 2 hours 15 minutes, Seoul to Gwangju in around 1 hour 40 minutes, and Seoul to Gyeongju in under 2 hours. Tickets can be booked online through Korail (korail.com) or at station counters and machines, which have English menus. Foreigners can also buy a Korail Pass for unlimited travel over a set number of days.
The slower Mugunghwa and ITX express trains are cheaper and stop at more towns. For budget travel, they are a good option where you are not pushed for time.
Buses
Intercity express buses are fast, cheap, and cover destinations that trains do not reach. They generally depart from a city's express bus terminal, which is usually near the train station or metro network. Tickets are sold at windows or from machines at the terminal. Booking ahead on busy holiday weekends is wise.
City buses within towns are cheap and extensive but can be difficult to navigate if you don't read Korean. The T-money transportation card (bought and topped up at any convenience store) works on buses, the metro, and even taxis in most cities — it also gives you a small discount on fares. Use it from day one.
Metro
Seoul has one of the best metro systems in the world: clean, punctual, cheap, and fully signed in English. Busan, Daegu, Daejeon, and Gwangju all have metro systems of their own. In Seoul, the metro takes you nearly everywhere you need to go, including out to the airport.
Taxis
Taxis are metered and reasonably priced by European and American standards. Standard taxis are white, silver, or orange; black "deluxe" taxis are more expensive. In smaller cities and towns, some drivers speak no English, so it helps to have your destination written in Korean or saved on your phone. Kakao T is the standard ride-hailing app and works well.
Domestic Flights
Flying within Korea is rarely worth it given the speed of the KTX. The main exception is Jeju Island, which is not connected by train or road. Budget airlines including Jeju Air, T'way, and Jin Air run frequent short-hop flights to Jeju from Seoul, Busan, and Daegu. Ferries from Mokpo and Wando are a slower but scenic alternative.
Romanization
A practical note: Korean place names appear in Roman letters in multiple spellings. Busan and Pusan are the same city; Gyeongju and Kyongju are the same; Daegu and Taegu are the same. Modern signage generally follows the Revised Romanization system, but older maps and books use McCune-Reischauer. A little flexibility and some time with Google Maps will sort out any confusion.