A menu for categories on a separate page

Hello! I have a grid theme and I can display my archives and posts of any category etc in 3-column grid.

Now I want to do away with my multilevel menu and replace my very long and dull drop down submenus by two or three pages with images and links orginized in the same 3-column grid. Of course visitors will have to click twice to get at the necessary page instead of clicking once but the submenus will look more interesting and engaging with big photoes and more eloquent descriptions. At least I want to give it a try.

One of my submenus I have ready: I created page-submenu1.php aand used the template for a static page and included it in my primary menu as one item. It was easy because all items in that submenu were links to separate static pages.

But how can I do the same for the categories on my blog? I want a page with images and descriptions of Category1, Category2, Category3 etc in a 3 column grid. And I want each image on the page to be a link to another page where exerptes of all the posts of one category will be displayed in a 3-column grid. I feel that I need some php coding but I don’t know which functions to use and in which files to put the code. Probably it ought to be another page template. Please give me some advice.