Getting one’s haircut in a foreign language is hard, especially when there is a swirling mass of overlapping vocabulary. To begin with, how does one even say hair? Chinese distinguishes between hair on one’s head and hair on one’s body. The first is 头发 (tou2fa, literally “head hair”) while the second is 毛 (mao2, “hair/feather/down”). Don’t get 发 (fa4 “hair”) confused with 发 (fa1 “send out”), the two characters are only the same because of simplification. Also, note that fa4 loses its tone in the 头发 construction.
To cut one’s hair, the operative word is 剪 (jian3 “scissors*”) and one could say 剪头发 (jian3 tou2fa). However, one doesn’t usually just cut one’s hair in China. It is pretty standard to get a 洗剪吹 (xi3 jian3 chui1 “wash, cut, blow”) where you get a shampoo first, haircut, rinse, and blow dry as a package deal. In smaller shops, you can save a buck by opting for a haircut only: 单剪 (dan1 jian3 “single cut”).
Gender also comes into play in Chinese between the pair of words 理发 (li3fa4 “tidy hair”) and 美发 (mei3fa4 “beautiful hair”), much like the distinction between a barbershop and a beauty salon. Both words can add a 师 (shi1 “master”) to the end to refer to the person holding the scissors (e.g. 理发师，美发师) or a 店 (dian4 “shop”) to the end to refer to the room where it happens (e.g. 理发店，美发店). Prices in salons tend to rely on the “experience level” of a hairdresser, and there is a whole lexicon of terms given them important sound titles, which I won’t go into here, because I always seek out the cheapest options.
In parts of China (even Beijing) where there are lots of old people still making up a community, you may find in public parks or on the street side an old barber with a pair of sheers. Give him a try.
*To refer to scissors, one needs to add 刀 (dao1; “knife”) to the end. By itself, 剪 typically functions as a verb meaning “to cut as scissors do,” so the full term 剪刀 (jian3dao1) could be literally thought of as “knife which cuts like scissors,” i.e. scissors.