By Rail
SJ (http://www.sj.se/) connects Värmland's cities with the rest of Sweden. On the website you can also buy tickets to Oslo in Norway, which is about three hours from Karlstad.
By plane
By plane
Karlstad Airport (http://www.lfv.se/templat...) has daily flights to Stockholm and Copenhagen. Tickets are purchased from the airline Skyways (http://www.skyways.se/). Another option is to fly to Oslo airport Gardermoen and go to Sweden from there.