Organizing posts into categories to display on certain Pages

Hello WordPress world;

I’ve seen questions close to this being answered but I’m still unsure how to make this work. I am setting up a wordpress site for my friend (http://theplaygroundkids.com) – I want to be able to make a post and choose which Page or subpage it will go under.

For example:

If I want to make a post under the blog, I’d just select the blog category and it would show up under blogs. If I wanted to post a research article, I would chose the news article category and it would show up under the News & Info > Articles/Quotes section.

I’ve tried creating categories but I can’t find a way to link them to pages and sub-pages. I also can’t seem to find a way to link pages and sup-pages to certain urls either.

Any help would be much appreciated, thank you!