Getting from Chiang Mai to Chiang Rai
 

Does any airline fly this route? Any other ideas for getting from one to the other?

I don't know of any flights. Hire a car and driver - it's about three hours drive.

There is a small airline the does this route once a day using a 12-seater Cessna aircraft. Expensive at 3800 baht. They are the same airline that flies Bangkok /Hua Hin. They also fly Chiang Mai/Pai.
http://www.sga.co.th/en-GB/index.html

If you have a day to spare you could take a bus or taxi to Thaton and then a boat to CR. Great scenery along the way.

You could rent a car and drive yourself. We did this in April, albeit flew from BKK to CR, and then drove from CR to the Golden Triangle. It was easy. I think the drive from CM to CR is about four hours. We reserved the car online from Budget.

Has anyone taken this flt between CM and CR? I'm thinking of taking it ow and it's reasonable, 1650 Baht.

Any idea how much a car and driver will run you?

Bus or taxi to Thaton and then a boat to CR sounds interesting.

Bus to Tha Thon and down the Mae Kok River is a good option. I did this about 10 years ago and found Tha Thon quiet and relaxing. I imagine it won't have changed too much. It's worth stopping for a day or two if you have time.

http://www.travelsinasia.com/Thailand/ThaThon.html