Yes, you are required to submit estimated tax payments as soon as you begin earning. Because you're self-employed, you'll also need to estimate self-employment taxes and submit that to the IRS as well.
The best resource for you at this point is IRS publication 1040-ES. There are worksheets in this publication (see page eight for estimating your quarterly payments). You'll also need the worksheet on page 6 to calculate your SE taxes due. I'm linking that IRS publication here for you.
https://www.irs.gov/pub/irs-pdf/f1040es.pdf
Good record keeping is imperative for you. Any qualified expenses associated with your income will decrease your tax liability.
You can make your estimated payments using several methods, including mailing with a tax voucher or paying electronically.
Best of luck on your new venture! Please post back if you have additional questions. When you are new to self-employment, it can be daunting. You'll get the hang of it!