Hi @treebeard
The easiest way to do this is to create a field group which contains the image field, and use the location rules to attach this field group to the taxonomy you want.
Now when you edit the taxonomy term you will be able to add an image. Thus associating an image with each term (cagtegory).
A page that shows all the taxonomies? That sounds like a taxonomy archive page and you can read about these on google / wp.org
