Here is a map of some of the many bike shops in Chiang Mai.
By pretty much universal acclaim, Yod at Four Heavens is the best mechanic in Chiang Mai, possibly in Thailand. His prices are very reasonable too.
If you’re looking for a good mechanic closer to town, I’d recommend the guy at Soi Tan in Santitham. Although not a shop and working from his garage, I’ve had excellent experiences whenever I’ve used him. As a bonus, his prices are very cheap.