Don't just stay in Seoul. My adopted hometown is Busan, and it is a city I absolutely LOVE. The atmosphere is much different from Seoul, but is still a big city ... but with the sea and seafood culture that is much different from Seoul.

I keep a public page on facebook for my friends. The link is <a href="https://www.facebook.com/HowardsInBusan?ref_type=bookmark">https://www.facebook.com/HowardsInBusan?ref_type=bookmark</a>;. Feel free to take a look. Like the page as I update it fairly frequently. Some things are from Seoul and Suwon since I work in those cities with some regularity, but since I live in Busan, most of the pictures on that page are from around the Busan area. I can tell you I will be going to Taejeongdae tomorrow to meet some friends we haven't seen since we lived in California (they're originally from Korea, and are back for a visit). Taejeongdae is a place I absolutely LOVE, and it is breathtakingly beautiful. I'll have pictures as soon as I process them after tomorrow's trip. Like the page, and it will give you updates when I post something new on there.

That's a lot of recommendations on one page, but pick out what interests you. I know not everybody is interested in teh same things. I'm a food, culture, and natural beauty guy myself, so that's what you'll see emphasized on my page.

Definitely, understand that there is much, MUCH more to Korea than just Seoul, though, and look outside of the Seoul area for some different interests and activities. :)

Thaddeus, in answer to your question, there is some monetary and time requirement to travel between the two. I make the trip between Busan and Seoul regularly for one day business trips, though.

If you take KTX, it takes about 2 hours and 45 minutes to get from Seoul Station to Busan Station. Be sure to look at the schedule, though because the trains that leave 10 minutes after the hour take a different route and take a little over 3 hours. It costs about 51,000 won currently one way.

So it's definitely doable for a weekend, but you are right that it takes a certain amount of time to get there, and I am adding that it will take about 100.00 USD to make the round trip via KTX.

@Steve: I totally agree.  저는 2011부터 2013까지 경상남도에서 일했어요. 부산은 서울보다 더 아름답고 재미있는것 같아요.

하지만, 시간이 많이 없으면 서울에 가기 더 편하죠?

Awesome, thanks a lot! 

I agree that activites in Seoul might be the easiest thing for you to consider.  Wonju's connection to Seoul is much better than with other cities.  In this weather, you might be thinking of spending time indoors, in which case you could consider looking around at Coex mall.  You could also check out the 63 Building.