I live in Suwon near Seoul and regularily travel there .
My favorite places have to be Jamsil or Insadong.
Insadong is a great place to see lots of art and to experience some korean culture and food but is a bit expensive.

Jamsil is a great spot for shopping or even for a cycle as you can rent bikes for free for a couple of hours and cycle around the beautiful parks here.In this are you have Lotte world a Disney style world and a really great Ice rink inside the Lotte Dept store there.

Myeondong is one of the best places to go for shopping with a large variety of stores and western type places for food .It is a nice place to go but very very crowded.

To see some art by some really well known artists check out the art Museum near Nambu bus terminal or the art museums located around city hall as I have already seen great exhibits showing the works of Rodin,Vangogh ,Renoir .

Seoul Grand park is another great place and has many interesting things to see .

If you are pressed for time and you want to see Seoul in a day, and have no personal guide, this would be helpful. The buses pass by a fixed route, and you can hop off at a place that interests you as you listen to their descriptions on the earpiece they provide you with - Korean National Museum, Changdokgung Palace, Insadong Street, Namsan Tower, Namdaemun, Dongdaemun, etc. -- and then hop on again at the designated waiting station, which are marked and easy to find. The tour gives a general idea about the city, with tidbits of history and culture. I suggest you start early, like 9 a.m. so you can take your time checking out each and every pitstop. 

You get to see almost everything! The best city tour I have ever had.