Background color

Take a look at the CSS in style.css line 708
Right now the color is #eeeeee. For white you would want #ffffff.
When you’re done it should look something like this:
#main {
background-color: #FFFFFF;
clear: both;
padding: 2em 0 0;
}

Question in general: is this best accomplished with a child theme or just go directly into the CSS?

Yikes… Make sure you or your server doesn’t update the theme automatically. Transfer your changes to a Child Theme or Custom CSS plugin before the update.

The Jetpack plugin has Custom CSS functionality;
http://jetpack.me/support/custom-css/
http://wordpress.org/extend/plugins/jetpack/