You can use the OWASP manual as a launch. It offers a set of tests that you can systematically use to check your application for common vulnerabilities.
Testing your web application pen is the sound word for what you are trying to achieve. Network scan for automatic tools and reference information.
Edit:
You mentioned that not only the client side is your problem, but also the general security of the entire application, including the server. My advice will be that if you have never done an app security assessment, your boss / site owner should probably consider hiring an outside company / consultant to work with. They will do the job less than it would probably be worth if you and your team had to learn the details first. In addition, they have the advantage that this is done over and over again, so they are much less likely to ignore important details.
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