Kyongju (often spelled Gyongju) contains a remarkable concentration of outstanding exapmples of Korean Buddhist art, in the form of sculptures, reliefs, pagodas, temples and palaces. Kyongju city and its surroundings inherited the glory that bloomed and withered in the ancient Silla Dynasty that lasted thousand years (57BC - 935AD). The city has been preserved through the designation of National Heritage and UNESCO World Heritage site.
Gyongju is a big, open-air museum. The center of town is easily walkable, with access to tombs, fortress walls, ruins and museums within a few minutes. Bikes are also available for rent. Mercifully, strict building codes have stopped those concrete high rise apartment blocks from taking over the center of town, and you see much more traditional construction in use.
The tomb park is worth a stroll, keep an eye out for the tomb that has been excavated! Its amazing, though slightly creepy, to be able to walk through a tunnel into the middle of one of those huge mounds. It is possible to see the construction, and some of the art found in the tomb.
Keep walking (or biking) and you'll come to and old tower which is supposedly the oldest still-standing observatory in east Asia. Keep walking and the path will pass an old forest where an old Korean legend is set. The signs tell the story in English, but the forest is nothing special. The fortress walls are up a hill, as usual, but they are mostly ruined. Nowadays locals and Korean tourists use the areas for activities.
You'll probably want to bus or bike to the museum, but don't miss it. It has an excellent collection of outdoor stone sculpture and inside an extensive collection of Silla artifacts and crowns. The huge bell in the front has a macrabe legend. The artisans failed again and again to properly cast such a big bell.
Then a fortune teller was consulted and they were told to throw the body of a young girl into the lead. This was done, and to this day when they ring the bell you can still hear the girl cry. I hope its just a legend! Take a look at the gift shop, it has a good selection of unique artistic items I've never seen elsewhere. Just across the road you'll see some reconstructed palace buildings and a nice, peaceful pond. It was built by an old king. Signs tell fun stories of royal drinking games played around the pond!
Don't bother with Bomun Lake. Most of the hotels are out there, but its nothing but a modern convention center far from the historic sites. There is a bus to town, but it isn't very frequent and takes quite a while. The lake path tends to be crowded even by Korean standards, so if you don't like being run down by children on bikes you can skip it. Oddly, the restaurant facilities are consistently facing away from the lake, so if you want to sit outside with a beer, your best bet is Family Mart!
There are extensive hiking trails all around. It would be impossible to choose one best trail, as there are varying difficulties and diverse sites on each trail. Just keep in mind that in Korea, unimproved REALLY means it, so wear good shoes and take plenty of water. Even the easy trails might have steep spots.