If your schedule permits stopping at only one city other than Seoul, that city should be Gyeongju, 165 mi/265 km southeast of Seoul. It was the capital of the Silla Kingdom (57 BC-AD 935) and has a high concentration of temples, statuary and burial mounds, as well as an excellent historical museum.
One of the highlights of your visit will be a walk through Tumuli Park, where you'll stroll among the burial mounds of 20 kings, some several stories high and looking more like hills covered with soft green grass than the royal tombs that they are. (Oddly enough, the park was only dedicated in the 1970s. Before that, the area was residential.) Just a few have been excavated, and the interior of only one is open to the public. Inside, an exhibit re-creates what it looked like when the mound was first explored, and some of the original contents are on display. Most of the finds, however, are exhibited in the Gyeongju National Museum, along with other artifacts from Gyeongju's golden age.
Outside of town is Bulguk Temple, one of the most beautiful temples in the country. It's a large complex with wonderful architecture, pagodas and statuary. There are other temples and mounds near Gyeongju (some have larger-than-life stone soldiers and bureaucrats to guard and guide the departed kings in the afterlife) as well as the Seogbinggo (where ice was kept long before refrigeration), Anapji Pond (a part of the royal pleasure garden), a very early star-gazing tower and other relics.
Also within easy driving distance are the Seokguram Grotto (containing an exquisite stone Buddha image surrounded by friezes) and the Poseokjeong Pavilion. In the mountains surrounding the city are thousands of Buddhist temples, inscriptions and pagodas (the roads to some are impassable without a four-wheel-drive vehicle). You could spend weeks driving around, but two or three days will give most travelers a taste of the area.
One other nearby attraction merits mention, though it is difficult to get to: the underwater tomb of King Munmu (AD 661-681). The tomb is submerged in shallow water in the middle of a little island about 300 ft/90 m offshore. Legends say that King Munmu asked to be buried there so he could return as a dragon to protect his people from invasion. Visitors must bargain with a local boat owner to get to the tomb.
If time permits, visit the village of Yangdong, just south of Pohang. Ignored by most tourists, the village seems immune to all forms of modernization.
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