Klarälvsbanan is a very good and level bike path to the north from Karlstad, in total 90 kilometers along the Klarälven river. It is a former railway track, so it is car-free. It is also very good for by roller scating.
Local and regional buses are managed by Värmlandstrafik. Schedules and a travel planner is available at their site Kollplatsen.
Safari. It's not Serengeti, but with a little luck you may encounter beavers, moose or even wolves. Wildlife safaris are organised from several towns.
Rafting on the river Klarälven is an interesting way of seeing the countryside. Slow wooden rafts, 4 by 4 meters, take you through the forests on 1-, 3- or 6-day trips. The trips start near Sysslebäck in the north of Värmland and go south from there. More info at the website of Sverigeflotten.