Gyeongju will be a nice place to travel for Korean culture as it used to be a capital of the Shilla dynasty.
They have some historical places there and the area is really open and nice to travel.
Andong near Gyeongju is another spot where u can go as well.
Suggest staying at a countryside Hanok residence in Andong where u can experience traditional Korean stay.
Something like
Nongam Jongtaek
https://www.tripadvisor.com/Hotel_Re...angbuk_do.html
This website seems to have a good writeup
https://www.creatrip.com/en/blog/10150
There is also a traditional Korean village at Suwon about an hour south of Seoul that u can visit.
Gangneung and Sokcho are beachside towns that might interest you.
Near Sokcho is Seoraksan mountain, quite popular in Korea.