Gyeongju is a city located in the Southern East part of South Korea. It is a 2-hour train journey from Andong. Gyeongju is one of the most important historical and cultural site of South Korea.
The city features several sights such as the Bulguksa Temple, the Yangdong Folk Village, several tombs spread across Gyeongju, the Gyeongju Historic Area, the Cheomseongdae Observatory, the Seokguram Grotto, the Donggung Palace & the Wolji Pond and the Gyeongju National Park. While the sights are countless, I do not think that any of these are must-see places.
Two days are good enough to get around the different sites – including a day of hiking the Geyongju National Park.