Air conditioned buses leave from Chiang Mai's Bus Station No. 2 and take about 4 hours. Thai International have daily flights from Bangkok which take about 50 minutes.
The most interesting way to travel from Chiang Mai is first take the early morning bus to Fang and then a songthaew to Tha Thon. From there take a boat down the Kok River to Chiang Rai. However, be prepared for a long journey (around 10 hours in total).
Chiang Rai is quite a small place so you can easily walk to destinations around the town centre. For longer journeys there are always plenty songthaews looking for customers.
Whilst in Chiang Rai it is worth visiting the Hill Tribe Foundation. They will tell you exactly where to find the Hill Tribe villages, which can save time, and they also have a constant supply of genuine items such as silver, embroidery, clothing etc for sale, but do not expect prices to be any lower. The Foundation is a non-profit making organisation and its mark-up is small. It pays high rates to the village craftsmen in an effort to maintain tribal income to a life-supporting level, in order that they do not return to opium poppy cultivation