Falun is about 3 hours from Stockholm, and is reached most easily by road or train.
By bus
There are very few bus connections from other cities. Swebus Express (http://www.swebusexpress.se/) has 1-2 connections a day with Stockholm and Gothenburg. Buses stop at the railway station which is a five minute walk from the city center and the hotels there. Unless your bags are very heavy, save the taxi money and walk instead.
By train
The train station is situated a five minutes walk from the city center and is served by the main train company SJ for connections to Stockholm. The local/regional connections are served by Tåg i Bergslagen/Tågkompaniet. Railway tickets can be bought at the SJ website (http://www.sj.se/) or Tågkompaniet only in Swedish (http://www.tagkompaniet.se/) All tickets are available on both sites.You have to be prepared to change train or to a bus from the city of Borlänge for the last few kilometers.
By plane
If you're in a hurry there is an airport at the neighboring city of Borlänge, about 20 minutes from Falun, serving Stockholm, Gothenburg, Malmö and Oslo in Norway.