There are 3 direct buses from Mo Chit bus terminal in Bangkok for Chiang Rai daily. Buses depart at 0750, 2000 and 2020. Journey time is 11 hours and fare vary between 450 and 900 baht dependent on class of bus.
There is no train service to Chiang Rai from Bangkok, head to Chiang Mai from here buses run very regularly to Chiang Rai. Bus fares start from around 60 baht. There are 5 overnight train service to Chiang Mai departing at 1430, 1800, 1920, 1935 and 2200. The journey to Chiang Mai takes around 12 - 14 hours dependent on service used, and the onward bus to Chiang Rai takes around 3 hours.