I haven't been there since 1987. Bill58 may see this and be able to comment on current transportation.
AFIK, the public transportation is still good. In the city of Seoul there is a subway system, and public buses. Both may be crowded from time to time. My second time there I used local buses from where I lived into Yong San. It was OK. You may want to consider buying a car. Will you be there as military, DA Civ, or contractor? That might determine if/how much it might cost. But many Koreans now have cars.
Outside Seoul, again there are buses, locals and highway buses. The latter can be comparable to highway buses in the USA. Trains are another way of moving around. I don't know what it is like now. Before, there was 1st Class (only from Seoul to Pusan, but I think that has expanded), 2nd class, and 3rd class. On 1st and 2nd class, you bought a ticket for a seat. 2nd Class was not as posh, but OK. 3rd class, you just bought a ticket for the train. If you were lucky enough, you got a seat before others did. If not, you stood or sat on your luggage in the aisleway. Some places only had 3rd class service.
There are lots of photogenic places. You will find some from talking to others, and some only by talking to Koreans. Or by asking USO or such places. Don't be afraid to stay in Yogwans, some are really hotels, but don't have to meet the same standards.
Good luck.