The well-paved roads around Chiang Mai are no problem for most drivers—even the mountainous Mae Sa route north of Chiang Mai is perfectly drivable. However, Thai drivers are notoriously reckless and accidents are frequent. Two major car-rental agencies in Chiang Mai are Avis and Hertz; Budget has a good range of four-wheel-drive vehicles. Many hotels have motorcycle rentals.
Avoid driving in the city during rush hours, which start as early as 7 in the morning and 3 in the afternoon, and pay attention to no-parking restrictions (usually from 9 am to noon and 3 pm to 6 pm). Parking is prohibited on many streets on alternate days, but the explanatory signs are mostly in Thai. Your best bet is to note on which side of the street vehicles are parking. Chiang Mai's traffic police clamp and tow away vehicles parked illegally. Parking lots are numerous and charge as little as B20 for all-day parking.
Hiring a driver with a car is the most convenient way to visit the hard-to-find temples outside the city. This option is expensive, though, and can start at B1,500 ($50) for a half day to B3,000 ($100) for a day trip. It is more affordable to drive yourself. Car-rental agencies also handle car-and-driver hires.