Train and bus travel to and in north Thailand an easy way to get around to most towns - buses go every hour or so from 0900 to 1600 approx so you never have to book or worry too much if you miss one. All buses have a conductor who keeps a good eye on everything so you feel safe even if you are the only visitor.

The train only currently comes as far North as Chiang Mai, there is also a sleeper service from Bangkok, but since the countryside is stunning you might prefer to take a daytime service to Chiang Mai and then the VIP bus to Chiang Rai.

If you are taking the train, you might want to consider breaking your journey for one night at Sukhothai and visit the historical park (think mini Angkor Wat) which is spread over several aquare miles, all completely flat with bicycle hire available to look around. 

Chiang Rai has 2 bus stations, the "old" bus station in town and the "new" bus station 7km to the South of town, just off the Superhighway.