changing background color without changing body color

Try adding this to the end of your child theme’s style.css:

#primary {
   background-color: #FFFFEE;
}

Adjust the color value to suit.

Sorry, I was a little hasty. You can change the body color to that value, and the #primary color to #FFFFFF, and that will change part of the page.

body { background-color: #FFFFEE; }
#primary, #secondary {
   background-color: #FFFFFF:
}

However, the #secondary area on the Home page is falling below the primary instead of beside it as in the normal case, so there is no good way to have a block of color for both #primary and #secondary on that page.

Also, I noticed that the link for Home in the menu points to a different location than the site URL. Apparently you have a Home Page, but have not set it as a ‘Static front page’ under Admin->Settings->Reading.