Best things to do: visit Nambu market by the river. It's a traditional Korean marketplace filled with just about anything you can imagine!
Also, try to be around during the month of May to see the Pungnamje, or the Korean traditional festival. You can make traditional paper, decorate fans, see very interesting and colorful street shows, and have an overall good time!
Quick Tips:
Spending time in JeonJu provides a great opportunity to try Korean food at its absolute best! JeonJu is the capital city of Jeollabukdo, which is the food province of South Korea. The veggies are fresher, the spices are just a bit spicier, and eating a good meal can be super cheap.
Best Way To Get Around:
The best way to travel around JeonJu is by taxi (and there are MANY!) or on foot. The JeonJucheon river stretches all the way through the city and has a very scenic and relaxing walking/biking path. Public transport is also very cheap (roughly about 70 cents) but the buses are not what one would call regular. They pretty much show up when they please.