Hi Brian :-)
I'm gonna visit Korea again during September, but slightly different days.
First, you should definitely buy a travel book, such as "lonely planet".
There are many things you probably want to visit, it's difficult to make suggestion for someone whose preferences I don't know ..
If you want to enjoy big city life and long nights with many drinks, stay in Seoul. If you want to eat great seafood, go to Busan or any other seaside city.
If you want to see rural and history places, go to Gongju, Gyeongju and Buyeo. Or maybe you are active and sportiv people = Diving in Jeju or hiking in Seoraksan?
I understand you visit on your own, or just two people. Make yourself a favor: Learn hangul. It won't take more than 3 hours, promise! and it's so much safer if you are able to read names on busses and coaches.
That with tickets also happens to me. This is my 6th visit to Korea and Japan after I stayed there for almost a year.. And it's my first time I'm able to visit a concert - AYAKA. :-)
I'm happy to help you with more questions :-)