There is almost no public transportation in Chiang Mai. There 2 main popular solutions are Sorng·taa (collective taxis), Tuk-tuk (individual 3 wheeler taxis):
For Sorng·taa you need to ask them if they go in the desired direction. If yes they will get you in and they can frequently stop to take other passengers. This is the most affordable solution. Tuk-tuk is a bit more expensive than collective taxis. You will also find them almost anywhere. You can also book regular taxis.