When I first arrived in South Korea, I mainly relied on taxis to get places, but using the subway and bus turned out to be more affordable and also more reliable.
In order to get around, people need to get a metro card called a "T-Money card" that allows you to pay for public transportation, including the subway, bus and taxi. In many cases, people have to use both the bus and train to get to their destination. While this might be complicated when first trying it out, the great thing about using the subway is that they announce each station when the train arrives in three different languages, including English.
Using public transportation was intimidating at first because there are so many different bus stations and train lines. After using it a couple of times, the process became easier. What helped me the most was using Naver Maps, a map application that gives a description of which line or bus to take as well as the transfer station.
