Enjoy your trip!
City: Seoul
Gyeongbokgung Palace
You might be lost in the palace so remember to take a picture of the palace map before exploring further.
There is a small admission fee which excludes the optional experience to dress up in a hanbok. Plus, free guided tours at various time slots in English, Japanese and Chinese languages. ^_^ Psst... Take note of the timing for their "changing of guards" ceremony!
Learn more about its history here.
Namsan Seoul Tower
Spectacular at night when the lights of the tower are switched on and the city lights shine amidst the blackness. It is a romantic place to be with your significant other. Bring a lock with you (or you could buy it there), write your names and lock it onto the same familiar fence that you see in Korean dramas.
Compared to other towers I visited (e.g. Eiffel Tower and Petronas Tower), Namsan Seoul Tower is not as impressive. They did however compensate their smaller observation deck with other activities. Other than the glimmering lights, there are also shops, restaurants and exhibitions. Visit their official site here.
Demilitarized Zone, DMZ & Panmunjeom
You will need to to book a tour beforehand for your trip if you would like to go to Panmunjeom (Joint Security Area). I went along with Seoul City Tour. The whole tour was smooth. Pickup will be at a hotel. It includes the 3rd Inflitration Tunnel, Freedom Bridge and lunch!
At Panmunjeom, there were 2 U.S. soldiers who accompanied my tour bus for security. Please be on good conduct and listen to their instructions. During my tour, there were some tourists who were stubborn and rowdy. This caused the more tension in an already tense environment. -.-"