The options you have for traveling within Mysore are buses which includes the Volvo buses, taxis and auto rickshaws.
If you are going to tourist spots then you should go for a Volvo bus which go to the major attractions like KRS Brindaven Garden, Zoo and Chamundi Hills. When travelling within the cit for shopping or food you can either take taxi which would cost you Rs9-Rs10 per km or take an auto rickshaw which should cost you around Rs11 per km, but most auto rickshaws don't go by the meter, so you would need to fixed the journey cost before hand and there may be some negotiation involved as well.
One other popular mode of transport in Mysore is the tonga which is a horse carriage, this may be a slow mode of transport but it is worth the experience at least once.