This sounds ideal to me.
The Four Rivers Trail, launched 2012, runs from the N.W. corner of South Korea to the S.E. corner spanning almost 600 km. Some claim Korea has the most advanced bike-touring infrastructure on Earth.
You can start from either Incheon or Seoul.
It’s mostly a flat paved route dedicated to cycling.
Read a 2013 trip report.
There are many other excellent long distance cycling paths in South Korea, as well.
Rick McCharles 
Hiking isn’t bad too, though a little different. We saw more cityscape and temples on our short hike in Busan. There were plenty of local hikers out there too and many seem to be on their daily walks.