Also, note that the Korean air conditioning situation is a tad on the weak side. So it's best to bring clothes that can be layered and unlayered as temperatures fluctuate. I am usually most comfortable in a skirt and top in Seoul and casual slacks, shorts or a split skirt outside of Seoul. For the guys, casual slacks are the best option. At the social service agencies, a dress shirt and a tie are in order.
Shorts are okay for men outside of Seoul and less okay in Seoul.
Comfortable walking shoes are a must for all. Also, if you are staying anywhere where you plan to swim, swim caps in Korean pools are treated a lot like they were in the U.S. 25 years ago. Many pools require them, and as sexist as it is, some pools require them only for girls and women. It is customary in Korea to take off your shoes as you enter many buildings and all homes. Slippers or scuffs are often provided upon entrance.