To any of yall who've been to Korea

I'm going in September with two friends and although I have an idea of areas I want to go, I have zero idea of specifics!!! I also have a friend who is born and raised there that will be taking us to Gyeongbukgong Palace and outdoor markets in Seoul. But I don't know how many days she's planning on hanging out with us/being our guide. 

SO if anyone has any suggestions for food spots/things to do/cute cafes that would be nice!!

I'll be in Seoul, then we plan to go to Jeju for a couple of days, and if time allows it Busan. 

A place near Seoul I'm dead set on going is Seoraksan Mountain too, since all of my Professors said it's a nice place to go to and the town by it is one of their hometowns!