I have a simple question about how to output a basic
"hello world" via one of the controller files to one of the template files.

So as a minimal case excercise, I have put

<p>{$myOutPut}</p>
in the appropriate place of the product.tpl file.

Now I want to pass stuff to that variable from the ProductController file.
I have created a new file in the overrides directory as such:
/overide/controller/ProductController

with ONLY this code.

class ProductController extends ProductControllerCore
{

    /// What the heck goes in here ???
        
}

Everything I've tried so far has produced either a total whiteout or rendered, partially rendered the page (top block, sideblock) or rendered the page completely BUT without the $myOutPut variable.

I have set the preferences/performance to force compile.
I am running PrestaShop 1.4

Could somebody give me just a tip in the right direction?

You might want to look at a complete though simple system - http://sourceforge.net/projects/templatecms/ is a fair start.