Do your homework
You probably have a favorite stylist and don't even know it. Take a look at the celebrities, photo shoots and people whose style you admire—you might notice a common name, and that's the stylist. Peep the credits for their name, and from there you can research how they started, and how to get in touch with them. Maybe that'll lead to a cup of coffee and some career advice, or even the chance to be their intern or assistant. You never know.