British citizens are able to enter South Korean without a visa for up to 90 days, providing they are visiting for tourism purposes only. You must be able to provide an onwards ticket upon arrival. Before you travel, you should apply for a K-ETA, which is a pre-authorisation to enter the country. This can be applied for online via the official website. If you plan to stay longer than 90 days, or should you want to work in the country, you will need to apply for a visa before you travel at the South Korean Embassy.
If you want to fly direct to Seoul from the United Kingdom, you will depart from Heathrow Airport (LHR). You can take a direct flight with Asiana Airlines, Korean Air, and Virgin Atlantic. The duration of these flights is around 12h 20min on average. It might be more convenient for you to fly indirect though. From Stansted Airport (STN), you can fly with Emirates, which transfers at Dubai Airport (DXB). From Manchester Airport (MAN) you could fly with Lufthansa, with a change at Frankfurt Airport (FRA). Alternatively, from Edinburgh Airport (EDI), you can fly with KLM, which requires one transfer at Amsterdam Airport (AMS).
For those that are planning to hire a car in Seoul, you will find there are numerous car hire agencies at Incheon International Airport (ICN). Many of the car hire desks, such as Avis, are located in the airport's Arrivals Hall. At the car hire office, you will need to show a valid driver’s licence and passport. In addition, you’ll need to have a credit card in the main driver’s name. It’s worth noting that some agencies will require an International Driving Permit (IDP), which you will need to obtain ahead of your trip from the UK.