Hi Rebecca,
I want to make sure we understand what you're wanting to accomplish, does this sound correct: When someone clicks on, for example, the "Family" page, they see first some content about the "Family" page which is persistent, then followed by Posts that are related to "Family", the Pets page would contain text about Pets, followed by Posts that are related to Pets, etc?
If so, what you want to do is use Posts (not Pages) and Post Categories (screenshot: http://i.imgur.com/ImbGrh5.png). Create a "Family" Post Category, then when you create Posts about the Family, check the box next to "Family" in the "Categories" meta-box to the right of the editor (screenshot: http://i.imgur.com/IXEcC0k.png).
Next, instead of linking to Pages in the menu, link to Post Categories (screenshot: http://i.imgur.com/RPT2b70.png).
The next step will depend on whether your theme supports Category Descriptions or not. To test this, edit the Category you created and add some text (screenshot: http://i.imgur.com/P8qqB2Z.png) then navigate to the new item in the menu to see if this is displayed to visitors or not.
If not, you can create a Post in the Category and make the Post "Sticky" (screenshot: http://i.imgur.com/1wgzmtS.png) - the theme *should* show the Sticky Post(s) first.
Another option would be to modify the theme output via a child-theme, which is more complicated but provides you complete control.
Thanks, and best of luck!