Seoul: what cool places are budget friendly?

I'm a US citizen planning travel in 26'. We are looking to spend 10 days in Seoul. I'd love to visit temples, palaces, & museums. Flights and hotel are covered. I'm already looking at subway / buses for transport. I'm not shopping for clothes. Clubs & bar scene is not for me.

I only want to spend money on cultural sites, amazing food, & skin care products (LOL).

Anyone have solid suggestions for places that are wonderful to experience without crushing ones wallet? Daytime & nighttime activities welcome.

Bonus: any suggestions for a luxurious high end dining experience? I want to go out for one really fantastic dinner. Basically: if you only had one luxury dining experience, where would YOU go?

THANKS