In the Secret World, I am aware of clothes shops in game but is there a hairdresser/barber shop style building where I can change my hair colour/style or any other place that allows me to change the appearance of my character?
Additional abuse of Google (and assistance from Shinrai) shows that there isn't a barber yet but that there are plans to add one sometime after the launch.
Running around the three major cities I have found a barber shop that is 'under construction':
So it would seem that yes there will be at some point. Looks like it will come in three parts: a barbershop, a tattooist, and some kind of appearance changing cosmetic surgeon of some description.