Seoul is a cosmopolitan and culturally fascinating city. As you walk around its streets, all the things that go to make up modern Korean culture are there to be observed.
Koreans are very fond of all kinds of food, and one particularly good place to visit is Noryangjin fish market: literally hundreds of stalls selling ready-to-eat fresh fish. In fact, you can buy some and have it cooked immediately in one of the restaurants in the area; or even eat it alive, Korean style, if you're brave enough!
Eating at a temple is another culinary experience you can have here. There are actually several temples in Seoul that are open to visitors and where you can not only learn something about the lives the monks lead, but also enjoy a meal (a completely vegetarian one of course) prepared by them and spend some time relaxing in a calm, peaceful environment conducive to mediation.
The family also plays a very important part in Korean culture, and weekends are often dedicated to family activities. In fact, if you want to have a relaxing time in a typically Korean way, you will greatly enjoy spending a day at a jjimjilbang. These are basically public bathhouses with hot tubs and dozens of rooms offering various wellness treatments (oxygen rooms, coal rooms, etc.). They serve as places for friends or family members to meet, remain open 24 hours a day, and even have dormitories you can stay in for a price that beats all the competition! It's worth trying for a night.
If you are more interested in immersing yourself in Korean culture however, the recommended thing to do is stay at one of the'hanoks': the traditional Korean houses found in the Jongno district. They normally have no more than five rooms to offer, and they usually have friendly, hospitable owners who will be delighted to teach you more about their culture, which still remains little known to the outside world even today! This is one of the essential experiences to have when visiting South Korea.
And just a little note to finish with: Seoul is simply overflowing with museums, but if you want something a little out of the ordinary, pay a visit to the world's only museum dedicated completely to Kimchi (a quintessentially Korean dish). It really is an unusual and unique place, as you'll see for yourself!