Eating in Korea was a bit of a challenge, primarily due to their main dish being Kimchi and other dishes that make you feel like your mouth is on fire. It's also hard eating while traveling alone whereas if you are in a group you can all order a bunch of things to share. Seoul was great as I stayed at Seoul Base Camp hostel and managed to save some money and cook a bit of food (you can spend all month eating a different type of instant noodle every day!). My high school lover Connie was living in Seoul at that time so she took me around to a couple places! One of my favorites was Mubanna Chondak. Not exactly traditional Korean food but dang! That chicken... mhmm. Places to stay away from KFC, I should have known better.
During my stay in Jeju Island, I didn't dine out often as I was part of the temple stay that included food. All the food that we ate were vegetarian and grown on the farm there. They days were I left the temple to do some exploring around the temple, I primarily ate little things here and there at food stands. They were all relatively inexpensive and so yummy!