Gyeongju was the capital of Silla, a kingdom that lasted from 57 BC until 935 AD. Its kings were buried under grass hills in the middle of town, a queen had a stone observatory built, and its monks carved a Buddha into a mountain. Here, thousand-year-old tombs stand between the coffee shops. The Busan and Seoul guides both send you here for a day, which gets you the famous sights and not much more.
The city itself is small and low, which makes it a place to walk slowly and stay overnight. The big sights are free or cheap, the best views are at dusk and after dark, and the places a day trip skips, like a folk village, a mountain full of carved Buddhas and the Silla palace pond lit at night, are the ones people remember. Three days is the right length: one for the tombs and the old town, one for the mountain temples, one for the villages and trails.
The plan keeps each day in one part of the region. Day one is downtown Gyeongju, day two is Tohamsan mountain and Bomun Lake, and day three is Yangdong village and Namsan. The guide also covers autumn colors, which peak here in late October and early November. If you want a wider look at the country, start with South Korea. Start on a Tuesday, Wednesday or Thursday: nothing important closes on Mondays, but Korean weekend crowds fill the tomb park and Hwangnidan-gil.