Visiting the Copper mine is a definite must! They have a tour that you can sign up for at the Visitor Centre/Gift Shop.
The tour takes about 1 hour and the guide will take you down to the mine to let you see where and how the miners used to work. It can get a little cold down there so bring a jacket along. You can easily spend half a day there if you can afford the time. There is also a museum if you are interested. Entry to the museum is free if you had signed up for the tour. Did not think that there were anything particularly interesting to shop in Falun as the name brands could be found pretty much anywhere. Nightlife was kind of dull when it turned dark. Otherwise, a walk around the town is also recommended.