The subway and buses are the main modes of transport in Seoul. The subway can get you most places, although the stops are further apart than on the tube so buses take you to the areas between. All public transport is paid for using the T-Money card, which is essentially the same as an oyster, and can be bought at the airport when you arrive. Both buses and the subway are cheap, espeically the subway when compared to the tube. You can plan your route using Kakao Maps, Naver Maps, or Cityplanner, although Naver is my personal favourite. Google Maps and other Western-made apps rarely work in Korea so generally the Naver (Korean Google) or Kakao-made apps are best. 

Rarely used unless along specific scenic routes where you can rent them cheaply for the day. Electric scooters are more common - you can use an app called Lime to rent them for however long you might need.

Very cheap compared to the UK and drivers are typically honest and trustworthy - just be wary of those going to and from the airport as there are lots of scams there and it's very expensive. You can hail cabs using the app Kakao Taxi which is the Korean version of Uber where you can pay in-app. If you hail the cab yourself, you can pay by cash, card, or using your T-Money card as mentioned earlier

Overground trains connect most cities of Korea and are overall pretty great! They are relatively inexpensive, but if you don't mind paying a little extra you can take the high-speed KTX train to a couple of major cities.