There are regular buses with Greenbus Thailand from Chiang Mai Arcade Bus station departing for Chiang Rai at least once per hour during the day. The journey takes approximately 3.5 hours along Highway 118. If you want to reach the city centre of Chiang Rai, get off at the Old bus station, the New bus station is on the outskirts of town.
Chiang Rai’s International Airport is served by various airlines (mainly domestic) with regular flights between Bangkok and Chiang Rai. In recent years, flights have started to operate from China and Hong Kong with multiple arrivals and departures per week.
If you’re interested to explore more than the city centre, it’s best to hire a car or motorbike to get around. Public transport consists mainly of tuk-tuks and red pickup trucks (“songtaew”), you can negotiate a daily rate to take you to various attractions.