Wander around Gyeongbok Palace
This is Korea's most well-known and primer royal palace. It was originally built in the late 14th Century, but a fire burned it down and left the space empty for nearly 3 centuries, until it was reconstructed in 1867.
Admission: Pay only 3,000 won ($3USD) to enter and you'll get access to explore everywhere, including the museum towards the back. The palace is enormous and there is a lot of interesting/cultural things to check out. Give yourself at least one or two hours to adventure around. It's location at Gwanghwamun Station is conveniently located next to Seoul's other tourist attractions, such as the famous King Sejong Statue and the Cheonggyecheon Stream.
Book Now