I may be trying to run before I can walk here, as I'm new to Joomla. But having gotten my site up and running in plain Joomla, I then started reading about K2 and decided I had to have it.

I missing the logic somewhere of how I actually switch. I have imported Joomla content into K2, and can see my categories and articles in K2, but of course the website is showing the old articles in plan Joomla.

I have, for example, a module called 'Products' which houses the category our-products, to display it on the front page. But it seems that I can't change the module to source articles from K2, so I guess there's a different way of doing it?

You probably have to change a lot, if not all, of your plugins and modules that uses your content. They have to be compatible with K2 content.

Most of these modules can be replicated with K2 Content.
If you have basic HTML/ CSS skills you can pretty much replicate all the functionality without the use of a 3rd party extension.