the han river has bicycle trails nearly over its entirety. for a short trip i don't think you'll need more than that. bicycle rentals should be available at several locations along the path - don't know specifically where as i have my own bike. :/
there is also a freeride community in seoul made of foreigners. they have rentals available but the riding is pretty hardcore stuff - granite faces going down at 45 degrees.
Korea is not the most bike friendly country. The roads are congested and most drivers are not looking for cyclists. Jeju-Do is pretty good. The topography will give you a workout. And there is the temperature extremes, hot/humid summers and cold/dry winters. Air polution and cyclones if that is not enough.
Enjoy your trip and get the workout indoors on trainers and such.
I am a Korean and live in Seoul. In Seoul, Han river bike path is 35km+ long one way and both side of river has bike path. I ride on that course almost everyday in the morning. They have bike rental booths operated by district offices.
I don't know anything about Changwon but the city is very much smaller than Seoul and might have better condition.
I think Jeju-do(the biggest island in Korea) is the best place for the bicycle ride. One round of Jeju-do is 250km and very scenic. I am sure you will enjoy Jeju-do very much. You can rent bikes in the island.
I will take my first bike trip in Jeju-do with my friends in November.
Information on bike tour and rental in Jeju-do: http://www.jejutopdong.com (in Korean, sorry)
FYI, Canadian dollar is 1170 Korean Won(KRW) as of 10/8/2008.
I lived in Toronto for 3 years from 1996 and visited Niagara falls 10 more times with my family and visitors from Korea.
I hope I can visit Niagara falls again and ride bicycle, do jogging and camping there.