Within areas like the Old City or Nimman, Chiang Mai is easy to explore on foot. To get to one district or another or to farther away temple and art galleries, though, you’ll want some kind of motorized transport.
The easiest, most hassle free option is to download the Grab app on your phone. Grab has recently bought out the Southeast Asian branch of Uber, so you don’t have many other options nowadays. A driver should be able to come right to you within minutes, and the rides are often surprisingly cheap, especially by taking advantage of discount codes.
Another easy option is by hopping on the red songthaews. These vehicles, which are converted pickup trucks, are a mix of private and public transport and are ubiquitous throughout the city. They ride around on normal routes, not unlike a bus, but will go slightly out of the way for you if you request it.
You just flag one down like you would a taxi, and tell the driver where you’re going. If he agrees, you hop in the back where you’ll likely encounter other passengers sharing the vehicle with you. Getting off, just pay the driver a flat fee of 30 baht.
Bear in mind, though, that if you approach a red songthaew that’s already parked, they will try to negotiate with you as a tuk tuk driver would. Speaking of tuk tuks, there’s no reason to ride them anymore now that Grab exists, as they nearly always try to quote foreigners outrageous rates.
As for regular private taxis in Chiang Mai, they do exist but are a very rare sight.
At the time of writing, the city has just recently revamped its bus system and has finally made the effort to translate some the bus stands into English. This would probably be the cheapest of all the options, but also the slowest.
Of course, renting your own motorbike is also an option. Be sure to have the proper licenses, as there are many police checkpoints all throughout the city.