Lighthouse in Chrome
Chrome Lighthouse is an open-source, automated tool for improving the quality of web pages. They got some really nice tests for Progressive Web Apps, Performance, Best practices and Accessibility (yay!). Lighthouse is available in developer tools in Chrome Canary and you can also install it as an extension in Chrome. The "best practice" is a little weird in my opinion but I'm guessing they, the people at Google, had a lot of things that didn't fit under any of the other labels so they had to come up with something?
When it comes to accessibility Lighthouse is powered by aXe-core and it tests for color contrast, correct element attributes, ARIA etc. I think it can offer a pretty good help when it comes to getting started with a11y testing.