Regular taxi from the airport to the old town or Nimmanhaemin area should be 180 baht, but from personal experience it could go up to 200 baht if you have lots of luggage
Grab is the Southeast Asian version of Uber, the price to Grab out of the airport is pretty much comparable to regular taxis (I had 180-190 baht) for Grab to and from the airport
There is a taxi counter you should go to just outside the international arrival (southern tip of the airport terminal building). You can try your luck and go up to one of the taxi drivers standing around in the area but the prices won't be that different
In Chiang Mai you won't be swarmed by taxi drivers trying to sell their services to you when you exit the luggage area, it's pretty quiet
For reference 180 baht is around $6. Exchange rate is approx 1 USD to 30 Baht