Google Search Console (known as Webmaster Tools) is one of the most important single tools to use for SEO. This is your best connection directly to Google and understanding how your website is crawled, indexed and generally interpreted by Google. You can use the Search Console to get alerts when there are problems, and to provide Google with important information about changes and updates to your content. It also serves as an incredibly valuable keyword tracking tool. (and it's free!)
Setting up for both www and non-www versions
Other sites and visitors will often link to your domain using both versions so it's important to capture both and set the preferred version so you can consolidate link equity and not have any duplicate content issues.
After you verify your www. version in the Search Console create a new property for the non-www. version. Since you've already verified the www. version the non-www will automatically have verification. Just click add property at the top of the main dashboard and follow the same steps as your www site.