10 days in South Korea, Wow!! First step, I reccommend you to see the general information of South Korea from here
http://www.asiarooms.com/travel-guide/south-korea/south-korea-overview/south- ...
then choose your destinations, normally 4-5 cities must fit for 10 days. Depend on your preference and your interest, South Korea is the country of traditional and modern attractions. Busan ,Seoul, Jeju Island and Haeinsa are the best destinations. Don't miss them.