Studio Theme - I would like to create a Category index on a Page.

I am trying to create a page that indexes a group of categories. Example I have a category of Recipes, with many sub-cats and want to have a page that allows a visitor to click on the category of their choice which opens up a page that lists the feature image and short description of posts in that category.

I thought I could create a template for a page and do something like this:
$cat_title = single_cat_title('display'=>false);
// THE LOOP ...
echo $cat_title;
But I don't know which template or php file to use as a master.

anyone help?