You need to install Google Analytics tracking code for a start. That will give you all the user behaviour you need , but you should also set up event tracking to track which buttons are clicked. You can also link these to goals where relevant so that you can measure conversation rates, but you also need this if you’re using Google AdWords/Ads.
There are other options for tracking but most are paid for and are costly, Google Analytics is free.
Also use HotJar or something similar for heat maps and recording sessions. This will give you huge visibility over what’s happening on your site.
These tools don't sell your data or pass it onto your competitors so it is kept private, only people with the login details to the tools I have suggested have access to the data.
The only alternative is you make your own version of these tools yourself!
Not using Google Analytics because they have access to your data is like not using an email provider because they'll have access to your emails. I think your position on keeping your data completely private will limit your options when it comes to SEO. As in SEO you will need to use tools like web crawlers, Heat map tools, Google Search Console, and a range of other off the shelf solutions to problems.