All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, we thank and praise Allaah Who guided you to Islam and we ask Him to make you firm and steadfast on His Religion until the Day you meet Him.
We advise you to seek beneficial knowledge, perform good deeds and befriend righteous and pious women, as these are the best things that help a Muslim woman remain steadfast on her religion.
As regards your question, according to some scholars it is permissible for a Muslim woman, when in the presence of a non-Muslim woman, to reveal any part of her body that usually appears when she does her housework, like the head, the arms and the neck. For more benefit, please refer to Fatwa 90940.
Therefore, it is permissible for a Muslim woman to go to a female hairdresser in order to do and style her hair even if the hairdresser is a non-Muslim woman provided the hairdresser’s is free from any religious prohibitions, like men entering there, or revealing the 'Awrah (parts of the body that must be covered in Islam), and the like. If these conditions are not fulfilled (which is most likely to be the case in non-Muslim countries), then it is not permissible to go there.
For more benefit, please refer to Fatwa 84749.