Pingdom will tell you how fast your website loads and how it compares to other websites, plus any current issues and potential fixes. At a minimum, you should keep the site loading time below 2.5 seconds and ideally, around 1.5 seconds.
PageSpeed Insights is a tool from Google which identifies improvements to make your website load faster. Google also provides additional website test services, including a test to determine if your site is mobile friendly.
Google Webmaster Tools shows any issues Google has detected about your website. You want to make sure NO errors are reported here.