Seoul is one of the best places to start. There are five Royal Palaces in this enormous city, as well as parks, shrines, markets and theme parks; it would take weeks to work your way around everything. One uniquely Korean destination is the Demilitarised Zone, or DMZ – something of a misnomer, given the amount of firepower that both North and South Korea have pointed at each other. You can see propaganda villages on the other side of the border, and visit tunnels that the Northern Koreans have dug underneath the boundary. Less alarmingly, you can visit Korea's many excellent beaches and pretty islands.
Busan is the second city and has similar attractions to Seoul, however it's particularly famous for its beaches and hot springs. Daejeon also has a fantastic hot springs resort, and there are a dozen high-tech ski resort areas with man-made guaranteed powder snow. Perhaps surprisingly, hiking is very popular with the locals with Seoraksan National Park with its spectacular cliffs a favourite – you can follow hiking trails into the mountains. Baseball and soccer also attract crowds, and you should definitely consider attending a match. South Korea is very stable, but there are occasional spats with North Korea – check the news before you go.