If you're going to have cellular data (which you should. See below): Grab is definitely the easiest and most convenient way to get to and from the airport in Chiang Mai. It's basically the Uber of Southeast Asia (Uber sold its Southeast Asian business to Grab earlier this year)
You need a working phone number to set up the app (they will text you a confirmation code), so if you plan to use it, download the app and set it up before you leave
Grab's rate to/from the airport is identical to regular taxi (180-190 baht depending on where you're going. ~USD$5-6). But I prefer it because 1) it's linked to your credit card so you don't have to worry about converting cash at the airport and 2) the driver can't play games with you with the fee (regular taxi drivers will do that sometimes)
There are promo codes you can use when you sign up so you get significant discounts and even free rides. This Twitter account and this Facebook page list all the current Grab promo codes for Thailand.
The promo code advertised at the airport is HELLOCNX, which apparently gets you 100 baht off for the first 2 times (new users only). A trip to the city is 180-190 baht so this is basically an over 50% discount
Another promo code I saw was TRAVELCNX. 100 baht off for the first 3 times (new users only)
If you don't plan to have cellular data: taxi is pretty much the only way to get to and from the airport. Even though there's supposed to be a flat 180-baht taxi charge from the airport, some drivers may tell you it's higher. The first time I landed in Chiang Mai I ended up paying 200 baht because no one was willing take me for less (I think because I had a lot of luggage). There are a lot of drivers loitering around the arrival hall so just approach them and ask/haggle for price. Have your destination name and address ready
A note about cellular data: you can buy a 4G-speed SIM card at the arrival hall here from the national telecom operator AIS and they will do all the set up for you so it's really hassle free. You just need to make sure your phone is not locked to your carrier at home. The cost is 249 baht (around $7.5USD) for a week of unlimited mobile data. For 15 days it's 499 baht and for 30 days it's 799 baht. I highly recommend having mobile data as it will help you immensely with getting around town