There are many places to eat and to see in Seoul. Here are some places, I'm recommending which are Insadong, SamChungDong and JongRo.
These 3 places are pretty much close distance and you can see many of Korean traditional tea, food and drinks restaurant as well. Also you can buy some Korean traditional souvenirs for your family and friends. I guess Sunday is better to visit there. almost 10 years ago
Years ago I got taken to a very tradition but also very amazing Korean restaurant where the large table quite literally was covered in dishes to try. It amply demonstrated the range of food in Korean culture. I don't remember the name of the place, but have asked a relative for the name.
Meantime, try this article http://www.cnngo.com/seoul/eat/10-best-korean-restaurants-seoul-114014 almost 7 years ago