I recommend heading to Sokcho and Seoraksan. I know you can stay in Seoraksan National Park but I stayed at a great little hotel called Seorak Morning Inn. It was roughly a 20 minute walk from the entrance to the park and the best rates we found (I shopped around all of Sokcho).
The express bus terminal sells tickets and runs pretty regularly all day to Sokcho.
You can read my blog post about Sokcho here (well worth an evening!) and about Seoraksan here!
In terms of trails they have a ton and the ranger station is very helpful just to the left of the main gate. They spoke pretty good English and have English maps. I ended up only having about 8 hours to be there so could only do the Ulsanbawi hike. Its ALOT of stairs but great exercise and provides a stunning view. Definitely take the cable car up to the "mountain fortress" and since you have more time I would try to do the main peak. I think its like an 8 hour hike each way. I know you can get permits to camp there but maybe not in the winter.
If you just want day hikes I recommend Bukhansan. You can take the subway to right there. I did two separate trails and there are some awesome views from all the peaks. 1st day in Bukhansan. 2nd day in Bukhansan. or Suwon Hwaesong (more of a fortress but still a fun hike and no the Seoul subway line)
Hope that helps! Let me know if you want any more specifics. I've got the trail maps somewhere in my desk still.
Edited by: LiveLearnTravelTeach