I heard Washington has no state taxes but I'm being asked to file quarterly to the Dept Of Revenue?

First off, I'm new to the USA so this will be my first time filing either state or federal taxes.

 

I'm a self-employed contractor so registered for a business license as a sole proprietor (was that the right thing to do?). To make matters more complicated, I am also about to start a part-time job on the side which will be taxed through the business I'm working for.

 

When I registered for my business license I started getting asked to submit what I thought were state taxes to Washington DoR on a quarterly basis, but now on TurboTax I'm hearing that WA doesn't have state income taxes so I'm wondering what DoR is asking me to file and whether I can somehow get out of it as a sole proprietor with no employees?

 

Can anyone offer any advice on making sense of all this?