Beauty salons are ubiquitous all over town, one of the biggest ambitions for many women is to run their own salon. Most are excellent and can fairly well handle non-Thai speaking customers. Problem is of course locating.
My suggestion would be to ask one of the hotel staff where you are staying where they would get their hair done, or washed, or set, whatever. Girls are likely to say why don't you use the hotel salon? I would tell her/them that the hotel salon is probably too expensive, and you are wanting to be able to get some authentic flavor of a good shop on the street and give your business to someone who needs it more. One or more of the girls might know a shop nearby (although most would frequent their own shops near their homes).
Similar for barbershops, though they are not as ubiquitous as hair salons. There are a few barbershops along Sukhumvit near the Landmark Hotel with lots of pretty girls wanting to cut your hair. For the most part, unless you are very particular, they are satisfactory. Hotels of course have good barbershops, but same caveats as with beauty salons.