From Seoul, take the KTX to Iksan, and then you can take a bus or taxi from there. You can also take the KTX all the way to the Jeonju Train Station, but sometimes you have to transfer in Iksan. This is the fastest but most expensive option, taking about 2 hours from Seoul. Express buses are available for as little as 15,000W and take 2 hours and 30 minutes.
From Incheon Airport, there are Limousine buses that run every half hour that will take you near city hall in Jeonju. It takes 3 hours and 30 minutes.
From Busan, express buses leave approximately every hour and cost 22,700. The trip takes around 3 hours and 20 minutes.
Every bus trip has at least one rest time during the journey for ~15 minutes at a rest area where you can eat something and use the restroom.
