Juno Hair is a well-known hair salon with more than 50 locations throughout Seoul and other areas. The Myeongdong locations have English-speaking stylists and are very popular in the area which has many tourists and foreigners. They do cuts and perms for both men and women and are up-to-date with the trends.
If you bring a picture of a specific hairstyle, it can help you and the hairstylists communicate to get what you want. Note that most locations besides Myeongdong and Hongdae will be unlikely to have English-speaking staff.