Korean public transport is often pretty good! Intercity travel is often done via coaches and trains, and city travel via buses or metro systems in larger cities.
Personally I tend to avoid trains for intercity travel due to how cramped they are, how busy they get, and how frustraiting the ticketing system is (you MUST buy a seat in the train, and if you want to sit next to someone you know and there's only individual seats available, well, better luck next time!)When we last visited Korea, we purchased a T-Money card. This is essentially a national oyster card where you can purchase and top them up at most convenience stores (such as the 7/11 at Incheon airport), then use them to ride on most public transport, from metro lines in Seoul and Busan, to buses in other cities. Me and Haeun would normally get our coach tickets from the kiosk in the airport, but apparently,
they also take T-Money cards as payment! So I would highly suggest people purchase one of these cards and top them up.