I have been coming to Korea since 1987, and boy has this country changed. I am a Korean-American, who has grown up with kimchi and chopsticks my whole life. My dad loves his native cuisine and my mom has been making home renditions of all his favorite foods since I can remember. I have now lived in Korea for over two years and even gotten married here. I used to visit every year up to eight times a year for work and pleasure since 1997 and developed a real taste for Korean food and even gotten to know new cuisines.
This is what I would do if I was visiting over a weekend and my recommended list of restaurants in Seoul:
1. Jja-jang myun. This is the Korean-Chinese food. Like every culture their own Chinese food has a different flair and Korea is no different. This pan-fried black bean sauce with bits of seafood or other protein is an instant craving. They do this noodle dish best in Korea. Every hotel will have this at their Chinese menu and it should be a good rendition since they are using top notch ingredients.
2. Kang-jang keh-jang. This is soy-based marinaded raw crab. They have a crab innard rice which goes well while you are sucking the raw crab from the quartered crab. You will need a bowl of white rice. You have to take the top shell and put in a bunch of white rice and use your chopstick to get every bit of goodness in the top shell and mix it in the mini crab shell bowl that is on the bottom of your marinaded soy. Really take a chopstick and get into that point to get the orange stuff out to mix with your rice. All is edible minus the shell.
3. Gob-chang. I would go for the old classic style in the Kyo-dae gob-chang, where intestines are rendered in their own fat with a mix of veggies and potatoes. Hit my articles on Gob-chang on how to order and eat.
4. Gom-tang/Hwe-jang-gook. This is the equivalent of that late night after drinking place that every place has it. If you are from NYC, it might be that Halal truck. In LA maybe your El taurino, but in Korea we got our Gom-tang/Hwe-jang-gook.
5. Jeju Black Pig. If you just purvey the markets of Seoul and you can tell that Korean pork is some of the best pork in the world. Jeju black pork can be some of the best pork in all of Asia. At the restaurant I am recommending you they will give you some sick cuts and as you watch the pork render over charcoal (a must), watch the fat drip in the salty, briny, hot sauce emulsion that they stick under the grill plate for the pork, but over the charcoal. Excuse the line and just grab a beer from your local mart while waiting for this Korean goodness.
6. Fried Chicken. KFC might have made chicken finger licking good, but Koreans make chicken knuckle biting awesome.
7. Street food.
Please note that the bbq kalbi places are missing here. If you come from a self respecting city or have internet access or a nice Korean mother, you can get this of pretty similar quality everywhere.