South towards Chiang Mai / Bangkok
Chiang Rai is a relatively small city by Thailand standards and it is possible to hitch almost from the centre as the main highway leading all the way to Bangkok passes right through the city. To eliminate city traffic though, it may be easier to head south on a local bus to The White Temple (Wat Rong Khun) which is about 13km south of Chiang Rai. Local buses heading towards Phan only cost 20 baht for a ride to the White Temple and from the main road here there is more traffic going a longer distance. The buses leave from the old bus station in the centre of Chiang Rai. Alternatively you can take a songathew from central Chiang Rai which can be bartered down to 30 baht per person.
For Chiang Mai you will need to split off from the main highway #1 at Ban San Pa Sak and head along highway #118, but a lot of longer distance traffic from Chiang Rai is heading to Chiang Mai anyway.