In the United States, we usually say hairdresser or hairstylist.

There is also what is called a "barbershop." This is a place where usually men go to get their hair cut by a barber.

At a "hair salon" (some people might say "haircutting place") men and women can have a haircut, or have different procedures done to their hair like shampooing, hair coloring, perms, etc. The people who work here are referred to as "hairdressers" or "hairstylists."

I would say "I'm going to the hairdresser".

hairdresser has many sinonyms

Barber,hair cutter,coiffeur

In my country, "hairdresser" is called "理发师"、“美发师”。

Thanks all !! :D