I have Drupal 7.14 and Zen 7.x-5.1. I cannot figure out how to simply change the background color. I have tried placing
background: red; in several of the css pages but have had no luck.
The Zen theme uses many different CSS files. You need to create a subtheme for Zen and then you are supposed to choose which layout you want. Fixed or Liquid.
Within the subtheme there are many CSS files. Depending on your choice of Fixed or Liquid you have to edit that css file...
layout-liquid.css or layout-fixed.css
Within that file you can change the parameters you are trying to override. As long as you don't have css files aggregated in the performance settings you do not need to clear the caches, otherwise you do.
To create a Zen subtheme you need to follow a tutorial: http://drupal.org/documentation/theme/zen
I recommend going through that completely and reading and learning as much about Zen as possible. One thing to know is that you never want to alter the core files of Zen, you only want to override them in your own subtheme.
Lastly, adding your own custom CSS file can be useful, but for this situation it would not be recommended because there are already CSS declarations for your use. I use a custom Zen subtheme for this site: https://www.fdanconia.com/ - I love the Zen theme structure and once you learn where everything is and how to override everything is a great theme to use.
First make sure in which file you need to make css changes.
Just go to Appearance > Settings links in Zen theme Now choose css that you are choosing
Fixed layout (layout-fixed.css)
Liquid layout (layout-liquid.css)
You can make the css changes in the file that you have chosen from this option.
Also still you need to firebug it to check which line you have to override for background change.
Hope it will help.