You could start the trip by taking the local bus from Chiang Mai -Chiang Rai city ( 3 hrs – 4hrs. ride). Then continue taking the local bus to Chiang Khong, the border of Thailand and Laos. Passing through the immigration process from Thai border and get in to Laos country (Huay Sai). From there, you could decide to take Laos local bus to Luang Prabang. Most people prefer taking the night bus as you could save your budget and reach Luang Prabang the next morning.

So….

Bus ticket from Chiang Mai – Chiang Rai is varies from 200 -350 THB/ticket

Bus ticket from Chiang Rai – Chiang Khong is 65THB/ticket ( Might be changed, you could check)

Overnight bus from Huay Sai, Laos – Luang Prabang is around 720 THB/ticket

Others would be your own expenses which you could plan and control by yourself.