What to do in south korea for short stay ?

There are plenty of place for sightseeing in Seoul.

Palaces, high rise buildings. - Google places to see in Seoul.

But I was staying longer, this is what I would do:

Don't bother with the Han River in Seoul. It's a vast, charmless waterfront crossed with ugly bridges and lined with utilitarian buildings. There aren't too many points of interest in Korea. It's really about the lively restaurants and particularly the cuisine, which is arguably the centrepiece of Korean culture. Do your homework beforehand. You only need to visit one buddhist temple. Take a trip into the countryside on a train as it's very inexpensive. Doesn't matter where, it's kind of all the same.

Depending on where you'll be, visit one of the palaces like Gyeongbuk. It is a taste of Korean culture and is really cheap (~₩5,000 or ₩3,000 if I'm remembering correctly).

Also, visit the Han River. Check for the light and water show at night as it is free to watch. Also, the Han has many little streams flowing into it that people can put their feet in and relax.

Visit Insadong or Nandaemun. Both have street markets for souvenirs and stuff but are also lined with restaurants and street vendors selling yummy Korean food.

Seoul has the most to offer with a few palaces right in the city. Namsan tower is fun. The giant markets are fun Namdaemun, Tongdaemun. Itaewon is a cool party spot with English spoken there.

There are some nice big parks too. Seoul National Park. I forgot the other one.

Visit a temple, such as Haeinsa

http://en.wikipedia.org/wiki/Haeinsa

if you like shopping- i suggest you got to insadong or myeongdong.

you can go to seoul to sightsee.