There is an express bus that leaves Incheon to Jeonju about every hour until the late evening.... No one is ever sure about the times but you can call 1330 and find out. It drops you off at the Core (pronounced Koa) hotel in Jeonju. Taxis are always waiting there.
You can also take this bus to Incheon from the Core hotel. The last bus is about 3am which is prefect for a morning flight.
the fare is about 22,000 won
The boat to Sonyudo leaves from Kunsan and takes about 2 hours. The schedule changes from day to day, so it's best to call ahead. the number for the Korean national tourist organization is 1330. you can get just about anyinformation about korea that you can imagine from this number.
generally there are about 2-3 boats a day... they fill up quickly on weekends and holidays so it is best to get your ticket early.
Kunsan is maybe 2 to 3 hours bus ride from Seoul