Okay, a couple of things to address here:

I'm trying to create a site with some (imho simple) functionality for it's users - you log in, set up when you're available on your calendar and other folks can browse those available hours and apply for a meeting which you then accept or reject. It would be a useful tool for my coaching division I'm working in.

This sort of functionality exists in some prominent services like http://www.setmore.com/ and https://youcanbook.me/. If you intend to build something that's a noticeable improvement of those, then great. Otherwise, I'd avoid reinventing the wheel and just use one of those.

The problem - my knowledge of programming is currently limited to simple, inanimate HTML websites and blogs from ready-made templates.

You're going to need to do some backend programming in something like PHP or Node.js to get this project done. You could dive right into it and learn the language with this project, but it'd probably go more smoothly if you do a Hello World tutorial and try to build something simpler like a to-do list first. Once you have a grasp of the basics in a particular language, you'll have a much easier time tackling this booking app.

From what I've been able to gather there are free and open-source CMS tools like GRAV, although flat-file CMS's aren't designed to handle such dynamic databases. So if one wants to do such things "properly", should you code up the database engine from scratch? Or are there available plugins that you can start from? I'd much prefer some ready-made skeletons that I could play with to see how they work.

A CMS platform isn't really appropriate for a web app. You have bigger problems to deal with than templating and presenting static content. Look into web app frameworks like Django, Laravel, or Express. You certainly don't want to do anything drastic like coding the "database engine" from scratch; a framework helps cover the basics like database communication and URL routing.