Depends on how long you plan to go for and how much you are willing to pay.
If you plan to go for one day, then your best options are:
Muju Ski Resort (about 3hrs by bus)
High 1 (about 4hrs by bus) http://www.high1.co.kr/eng/main/index.asp
Muju is the most southerly and when it has good snowfall can be a fine resort.
High1 is of the newer resorts and offers a good weekend worth of gliding.
If you are willing to travel a bit further, then two I can recommend are:
Yongpyong (6 hours by bus) http://www.yongpyong.co.kr/eng/ski/overview.asp
Phoenix Park (5 hours by bus) http://www.phoenixpark.co.kr/english/intro_resort.html
Yongpyong is probably Korea's best resort because it has the most extensive range of slopes. Though it can take some time to reach from Busan, it is well worth the extra journey time. Also, if you plan to stay , the hostel is cheap and just across from the slopes.
Phoenix Park is not the biggest resort, but the fast lifts and compactness of the resort makes it a good for a one or two day trip.
I mention bus times because, unless you have have a car, the only way to reach these resorts is by bus. In order to book yourself on a bus you are likely to need some help from a Korean friend. One reliable operator is:
Plus, if you book your bus ticket through them, you can get heavily reduced lift passes. For one day trips they usually leave from various points across the city at about 2 or 3 in the morning and arrive at the resort by about 8:30. The bus then leaves 4:30 sharp in the afternoon same day.
The cost is about 50-70,000 for the bus, then about 30-60,000 for the lift pass.
A plus side for staying the night is that all these resorts have good night slopes (at extra cost).
All the resorts provide equipment hire.
Expanding on what absolute says. We used saepusan a couple of weeks ago. They provided a great service . It cost 82,000 for morning and afternnon skiing (8.30- 4.30) bus fare is 30,000 + 52, 000 for lift and equipment rental. They picked us up from Namcheon Megamart, but there are countless routes all over the city between 4-6 am. Taking about 3 hours to get there. This exact link gives you all the Muju details.
A lot of people criticize Muju as the worse of the bunch. It does have the longest and highest, but can get pretty damn busy in the afternoon. If you've seen the roads here imagine that with Kids!
Hi One total package is 90 with buses leaving at 2-3 a.m and about 4.5 hours. I have never been, apparently it has a casino, if you were planning a couple of days it might make things interesting. All of these resorts have a lot of condos and pensions etc. That also have good package deals if you want to do that.
Phoenix is regarded as the best place for snowboarding. Wheras the other 2 resorts above charge an extra 2 bucks for snowboard. Phoenix is more. The total one day package for skiing is 129,000. Wheras snowboarding costs you 153,000! Buses are similar to that of High One.
If you have basic Korean, or a Korean friend Saepusan can easeily be booked online, you just require an alien card number. If not give them a call. The tour guide on the bus we took had great English, so you should be able to find someone speaking English somewhere. (051) 851-0600 or 711-6000