It's pretty useful to get an outside objective look at your site and what deficiencies it might have that you can work on to improve. I use the following sites generally.
This is the premier site for checking the health of your SSL Certificates. Are the allowed encryptions up to scratch? The higher standards you have the more users may potentially not be able to access it due to old browser. As a rule of thumb, I expect all my sites to have at least a A score on SSLLabs.
Pingdoms speed tool
There are several sites that provide analysis of your site and what makes it
slow. This is my favorite, since it provides a broad and deep analysis. You can easily see the key benchmarks, such as number of resources loaded as well as a complete breakdown of how the site functioned and how well it follows speed recommendations.
Erics Pony Checkup
A tool specific to my most used framework, Django, it provides a quick and simple examination of whether you site follows recommended security advice. It's pretty high level and only checks 10 things. However, it is very approachable and the few things it does check are pretty important ones for your applications security and not many other tools like it exists.
Chromes Audit tool
The Chrome browser comes with a surprisingly great tool for analysing websites, with a focus on the performence and accessibility seen from the perspective of mobile users. It looks at first meaningful paint for instance, which is a very important criteria for slow connection users.
Some of the hardest deficencies to find are those that that do not affect yourselve. It's very easy to be blind to others problems. Luckily there is a great site that checks how accessible your site is and provides useful pointers on how to make it better.