Is there an easy way to adjust the block layout? I have menus that are set to Sidebar First in the Bartik theme.  When the screen size decreases, those menus are shown below the main page content.  Which file in a copied version of the Bartik theme could I play around with to change around the block configuration as shown here:
The page.tpl.php file within /themes/bartik/templates/ folder is the main layout for the theme.

So in your copied version of that you can edit that file. If you look for:
print render($page['sidebar_second']);

That is the rendered content of sidebar second region. There's also 1 for sidebar_first. So you can move these to a different location within this file if you want it to be displayed heigher/lower in the content