Always book ahead as Koreans love to travel around the country and space is limited. Any travel agent can do this, or you can do it in person at the station. There are inner city local buses going most anywhere in country and Express buses that service the larger cities. These buses have two classes neither class has bathrooms for some bizarre reason. I guess it cuts down on drinking.
Buses are cheap.
Seoul and Pusan have great subway systems that cost on average 600 won. Both these methods are fast and frequent. Your institute will show you around your new city and teach you how to use the local transit systems. Taxi's in Korea are cheaper than here, and you never have to tip. However, cabs will pick up other people heading in your direction to fill the car.