There's no train connection with Chiang Mai Airport.
Car and bus
Chiang Mai Airport is situated 4 kilometers southwest of the city center. The airport is easily accessible by rental car via the so-called Super Highway (for which you have to pay toll), or via the road from Chiang Mai to Hod.
A bus runs every 20 minutes between Chiang Mai Airport and the city center; costs: 25 baht.
Car rental Chiang Mai Airport
You can find counters of car rental companies in the Arrivals Hall. There are several rental companies present at Chiang Mai Airport, including local providers and international ones such as Avis and Hertz. You can also book a rental car in advance using the service provided by EasyTerra; they compare the rates of several providers in a single quote.
There's a counter in the Arrivals Hall where you can book and pay for a taxi. There are set rates. A taxi ride to Chiang Mai's center costs 120 baht. So-called songtheaws (a sort of truck with in the back a small seat on both sides) and tuk-tuks aren't allowed at the airport, but are available on the public road, past the parking lot.
There's a parking lot with space for approximately 500 cars in front of the terminal. There are no separate short or long-term parking places. You pay 10 baht per hour.