Categories side menu – how to expand into post titles?

Hey all,
I recently got the latest WordPress and started learning how to use it, but Im not very familiar with PHP.

I managed to install a theme I liked and edit some of the PHP to get the layout I wanted. I have made some posts and added them to categories. I expected that when I clicked a category type it would expand into a list of post titles. However, it just takes me to a page of all the posts in that category.

My problem is that my site is a resource site and I want people to be able to flip through and find old stuff easily, as its just as valuable as new stuff. But its not effective to have a random list of resources spread across several pages within the category, I need a menu within each category.

I tried to install few plugins, but I wasn’t too sure where to add PHP tags and I couldnt get them to work.

I want the simplest solution. I searched through old posts but could not find a working solution. Id appreciate any advice you guys could give me.