From Chiang Mai(Rai) to Luang Prabang on bus
The distance between Chiang Mai and Luang Prabang is only 750km. But due to a border crossing and bad road conditions, It can take up to 24 hours.
Bus to Chiang Rai
The first destination is Chiang Rai. Bus to Chiang Rai goes from the third terminal. I recommend buying tickets online at Green bus because when I first came to the station, there were tickets only to 6 pm bus.
To Lao Border
There only one bus station in Chiang Rai. The bus to the border departure from it from 6 am to 5 pm. The journey takes about 3 hours and costs just 100 baht. Before the turn to the border, they ask people if they need to get there. There is a took-tooker who will drive you to the border for an extra 60 baht.
At the border
At the border, there is a money exchange on both sides with a fine exchange rate. After checking the passport there will be a bus, which cost 20 more bahts, remember this if you decide to change all Thai money at an exchange.
The bus crosses the Mekong river, it is only 5-10 minutes to the other side. On another side, it is required to do all the typical stuff on crossing border.
A russian citizen gets the stamp. But most tourists need to pay 35$ for a visa.
After the border, there will be a small bus which can ride to the bus station.
Huay Xai bus station.
A bus to Luang Prabang is going at 6 pm every day. I got to the station at about 1 pm and could buy a ticket. It is better to buy come to the station in advance to buy the ticket because later there were about 3-4 people who didn't get a place on the bus and spend a night in an aisle.
The ride was about 14 hours long. Obviously, it is not an easy ride and it is very long. Bus stop at some station to get locals out but there is no shop at this station. About 9 am bus came to Luang Prabang bus station, from which took-took rides to a hotel/hostel.
It is a long but cheap ride. If you have time and not much many there is no problem in traveling from Chaing Rai(Mai) to Luang Prabang on a bus.