As mosburger mentioned, Seoul doesn't really have much sightseeing. I visit to say, Kyeong-ju would be much better for cultural sightseeing. I don't know what to call it but Seoul's more of a "lifestyle" tourist place. I think the best experience you can receive in Seoul is to take in the city like a local person would.

As for sights, I would go to Cheonggyecheon. In downtown, there are the various gates such as Gwanghwamun and then there's Duksoogoong palace. There's Insa-dong which is kind of an "older traditional Korean" district in Seoul. All of these places are within walking distance of each other.

For shopping, there's Namdaemun and Dongdaemun. If you're not really into the shopping but just want to check it out for cultural reasons, I'd choose Namdaemun but if you actually want to purchase clothes, I would choose Dongdaemun.

Almost every place in Seoul is accessible by subway/foot and if someplace isn't, then it'd only be a short cab ride (few dollars at most) from a nearby subway station.