Please recommend winter preparations and free activities in Seoul.

Hello. I will be moving to Seoul this December, and I have many concerns. Since this is my first time moving abroad, I have a lot of questions. I'm particularly worried about preparing for winter. I have always lived in hot countries, so I don't have any clothes suitable for sub-zero temperatures. Is there a good place in Korea to buy high-quality winter clothing? Also, to save on my budget, I am looking for activities that I can enjoy for free or at a low cost without working for a while. I enjoy activities like visiting museums or hiking, so are there any recommended places in Seoul? I would appreciate any advice. Thank you! <3