Getting there
Air
The main gateway into South Korea is the super-efficient Incheon International Airport. Dozens of airlines fly in and out of South Korea, so you shouldn’t have a problem finding a flight there.
The national carrier is Korean Air, which has flights right around the world. Asiana is a Korean-based airline that flies to the US, Australia, Europe and across Asia.
Incheon International Airport is about one hour from Seoul. There are regular buses from the airport to all parts of central Seoul. The helpful English-speaking staff at information points in the airport will be able to direct you.
There are several other international airports, but they mainly just do flights to and from Japan.
Sea
There are ferries from Korea to Japan, China and Russia. While they are a viable option for getting in and out of Korea, they are overly frequent, so do plenty of research beforehand if you are on a tight schedule.
Getting around
South Korea’s public transport system is excellent.
Bus
Buses link every town and city in South Korea, and run frequently. Getting to outlying areas or national parks by bus is straightforward, and most buses run hourly.
Train
South Korea’s train network is excellent, and the best option for cross-country journeys. In Seoul, the subway system is extensive and easy to navigate, and the trains run frequently.
Air
There are domestic flights in South Korea (the longest flight time is about an hour) but the country is small enough and the public transport is efficient enough to make them unnecessary for most travellers.
However, to get from Seoul to the island of Jejudo, situated of the south coast, a flight could be a good option.
For domestic flights try Korean Air and Asiana.
Jeju Air is a low-cost Korean airline. However, their webpage is only in Korean, so you might need help from a travel agent.