I just went last Christmas to Germany on a 'Christkindlmarket Tour'. The cities included Rothenburg ob der Tauber, Nuernberg, Bamberg, Salzburg, and Munich.
Each Christmas market was unique and I found things at each that I would not have found anywhere else. Nuernberg has the largest Christmas market in Germany.
For the time period you're going I think all the regular sights (museums and such) would be open.
Rothenburg is just magical at that time of year. Quaint shops, a small market, a crime/punishment museum... I would definitely go there.
Nuernberg is a bigger city and has a large pedestrian zoned area of major shops. Lots of history there too.
In Munich we did a city tour. Lots of great museums, the BMW museum & factory, Dachau concentration camp. The Marienplatz has a large Christmas market that is wonderful.
Salzburg is my favorite town, but I love Mozart and 'The Sound of Music'. You can see attractions for both plus the Salzburg cathedral, salt mine tour, funicular to the fortress, and listen to classical music almost every night. They also have a wonderful Advents Concert at one of the local churches there sometime mid-December that is just awesome.