I started my own business as a hairstylist and was told I need to register my business and pay quarterly taxes but I'm not sure how to do that or how much to pay

You can register your business with the State, County or City that you reside. Contact your appropriate municipality for more information.

In order to pay quarterly taxes, you will need to sign up with the IRS. Here are three links that explain more about quarterly tax payments:

http://blog.turbotax.intuit.com/tax-tips/dont-forget-to-pay-your-estimated-taxes-3637/
https://www.irs.gov/individuals/self-employed#QuarterlyPayments
https://turbotax.intuit.com/tax-tools/tax-tips/Small-Business-Taxes/Estimated-Taxes--How-to-Determine-What-to-Pay-and-When/INF12007.html