>>656652nd this, pretty much hits it on the nail on everything you should do. Went to Seoul a couple years ago for a week and had a blast. All signs are in English so it is easy to get around on the Subways. I didn't get any backlash for being an American. Downtown Seoul is fairly polluted so beware of the heavy smog.
If you get bored, you can go downtown to jump on a supertrain for $40-$80 and go anywhere in South Korea in a few hours.
To add, go check out Olympic Park for an hour. Lotte World is a joke compared to Six Flags. Techno-Mart with all the floors of stores albeit overpriced. I didn't get time to go check out the DMC but if I went back that's where I would go first.
The food is amazing (try bulgogi) if you're into Korean food, Sujo is super cheap too.
If you're into shopping, Seoul is awesome. Everything is setup in districts so if you want to shop for clothes, you head on down to Namdaemun Market. Electronics, Yongsan. 2nd hand stuff, Hwanghak-dong. When you go to these districts you'll notice that shops or even entire floors sell one particular type of items. If you want video games, you'll find a row of stores. Want video projectors? You'll find 8 A/V companies lined next to each other.