Having been in Chiang Rai a few months I quickly realised that the bar scene revolving around P4P and bar girls is quite limited. But it does exist.
The main bar area for this sort of entertainment is in the city centre along Jetyod Road, close to the famous Chiang Rai clocktower. There is a mix of 'normal bars' and P4P bars stretching along about 250m of the road.
Naturally, there are more P4P girls in high season than low season.
Everything is low key and not 'in your face' as it is in some areas of Bangkok. The bars are normally quiet until late into the evening, with most P4P girls coming to work after 6 or 7 pm. Depending on the level of business, some bars stay open most of the night behind closed shutters, but normal closing time is 12 midnight.
Some bars have pool tables, some don't. A couple have live music on occasion. But all in all a farang can have a pleasant time in Jetyod Road as long as he or she doesn't expect too much - unless they ask (and pay).
But before asking get the feel for the bar first, as not all bars in Jetyod Road have P4P available !
By the way, there are zero 'a go go' bars that I have found in Chiang Rai. Apparently there used to be but they were closed down a few years ago. I have heard that there is a Thai style go go bar somewhere in the city but I don't know if this is a myth or real.