Korea in 2–3 days
In two to three days you can fit in quite a few sights and experiences of Seoul, and a day trip to the demilitarised zone (DMZ).
The DMZ is a must-see, and numerous companies offer day trips there.
While in Seoul, make sure you have a Korean barbecue and give yourself time to experience a Korean bathhouse.
Korea in 7–9 days
This is a good amount of time to spend in South Korea.
It would be worth devoting three or four days to Seoul, and make sure you take a day trip to the DMZ. Then head to the interesting coastal town of Sokcho, from where you can visit the stunning Seoraksan National Park.
Korea in two weeks plus
You can see quite a lot of South Korea in two weeks or more.
After Seoul and a trip to the DMZ, you could head to Sokcho on the east coast, then travel down slowly towards Busan.
You could take a trip out to the rugged, unspoilt Ulleungdo island (off the east coast), or the equally spectacular Jejudo Island (of the south coast) where there are plenty of activities to fill in at least four days.