If you have the opportunity to travel to Innsbruck at the end of November until some days before the 24th of December, you can enjoy the Christmas market. There is one in the Old Town at the Golden Roof (see picture) and one at the Landhaus square.
Enjoy the atmosphere with hot Glühwein (mulled wine) and local specialties, although they are getting more and more expensive each year.
You can buy any kind of souvenirs, especially Christmas gifts, nativity sets and figures.
For me, a "real" Christmas market experience is, when you are freezing in your toes and fingers, but the Glühwein and a good conversation with your friends warm you up :-)
Oh yes, one more thing: note that Christmas in Austria is a very quiet, nearly meditative celebration. Celebrations take place in the closest circle of the family, people do NOT go out on the evening of the 24th of December and there are hardly any public celebrations. One or the other hotel will decorate a Christmas tree and organise a dinner for their guests, but that's all. You will have a hard time to "celebrate" Christmas as you might be used to in England for example. That is why some of the people might be disappointed if they travel to Austria in Christmas time wanting to have a "good time" in that meaning (I saw some forum entries, that's why I thought I might clear up that misunderstanding).
It's wonderful how the people of Innsbruck celebrate their new year's eve. There's a big party on the streets of the old town. At midnight there is a big firework display at the river - along with music especially composed for this event. Afterwards the people are waltzing in the street. A very friendly and relaxed atmosphere. I can really recommend this place for new year's!