Public transport consists mainly of tuk-tuks and songthaews, plus a smaller number of taxis. The city itself can be explored on foot, but for trips into the province consider renting a car or a motorbike. Budget and Avis both have offices in Chiang Rai and there are several local agencies. If you don't feel up to driving in what may be a very different environment, then consider hiring a car with driver - it doesn't cost much more.
Chiang Rai has a few cyclo-rickshaws, which can be a pleasant way to see the main sights if you want a fairly quick tour.
