Seoul, Busan, and the towns in between do different jobs, and a route that treats them as one long list of sights ends up a blur of palaces. Give each place a role, and the country starts to make sense.
Seoul: Palaces and the First Full Day
Almost every route begins in Seoul, and the full-day city pattern is predictable enough to plan around. A Seoul tour usually opens at Jogyesa temple, moves to Gyeongbokgung Palace for the Royal Guard Changing Ceremony, then covers the National Folk Museum before heading out of town.
One detail catches people off guard: on Tuesdays the palace stop becomes Deoksu Palace instead, so if Gyeongbokgung is the reason you came, do not book that day. The Korean Folk Village sits about an hour and a half from the city and holds around 260 types of traditional houses, which fills an afternoon on its own.

Busan: Coast, Beach and Cable Car
Busan is the counterweight to the capital, and most of what you came for sits along the water. Haeundae Beach and Dongbaek Park take a couple of hours, the Songdo Sea Cable Car puts you over the sea rather than beside it, and the UN Memorial Cemetery and Nurimaru APEC House round out a day without much driving.
A Busan tour works well as the halfway point of a loop, since the city sits at the bottom of the country and the routes back north usually run through Gyeongju.

Gyeongju: Tombs and Observatory
Gyeongju holds the Silla-era sights, and they sit close together. The Daereungwon Tomb Complex is a group of grassy burial mounds you walk between, the Cheomseongdae Observatory is a short stop that takes about twenty minutes, and both pair naturally with the temples outside town.
Jeonju and Andong: Hanok Villages
Jeonju Hanok Village keeps around 700 traditional houses, and wandering the lanes on foot is the point rather than any single building. Gyeonggijeon Shrine is in the same town and takes about an hour. Andong Hahoe Folk Village is the other half of the pair: it is UNESCO-listed, people still live there, and Queen Elizabeth II visited it, which is the kind of detail local guides enjoy repeating.