Daegu is the 3rd largest city of South Korea. It is located in the Southern part of the country. It is a short 1-hour bus journey from Gyeongju. While Daegu is not a tourist hotspot on the classic South Korean travel itinerary, I believe it is much better than Gyeongju.
There a few places of worship, street arts and parks to explore across the city. There is a nice 12km hike starting from the Donghwa Temple. The first section is a 3.6km path with a positive 671m from the Donghwa Temple to the Bilobong Peak which stands at 1193m above sea level (Palgongsan). The second section is a 7.2 km path with a negative 379m from the Bilobong Peak to the Gatbawi Stone. The third and last section is a 1.4km path with a negative 418m from the Gatbawi Stone to the Gasbawigil Road. Even though it is only 12km, it is a quite intense hike.
Two days are good enough to get around the different sites.