Below is my list of go-to links for website testing.
I test a website in these first and make improvements based on the information returned.

W3 Schools Validator: HTML code validator.
This checks the code, it catches unclosed tags and other errors that might cause problems in other browsers or further down the line.

Webaim: Checks the accessibility of a website
This checks a site behind the scenes. It looks at how the site would be perceived by a screen reader and how the site performs for those only using the keyboard to navigate. It also looks at colour contrast.

Nibbler: This test gives alot of useful info on a site and tips on how to make improvements.