To start here is a link on tips for travelers (be sure to get a cellphone at the airport.)
And here are some restaurants I would recommend. Here is an Eel restaurant. http://www.seouleats.com/2008/
And you can't forget about the famous Noryrangjin Fish Market
Oh, and Great Korean Bulgogi: Sariwon.
Myeongdong Calguksu is great for amazing noodle soup. It's better than Momofuko
Star Chef is a great place and run by a very popular chef.
And you can't forget about Korean fried chicken...yum
For high end there's Pierre Gagnaire, Wooriga, Jungshikdang, the Shilla Hotel, the W-Hotel, and the Intercontinental.
I'd be happy to show you around. Also, I know some cooking classes as well.
My nemesis Roboseyo also did an excellent post on Korean restaurants. He even marked things on a map. http://roboseyo.blogspot.com/2009/01/just-because-i-love-to-share-get-fat-in.html
Oh, and here is another post I wrote for another tourist.