While the earth is orbiting the sun, I would like to make the moon orbit around the center of the earth.
Is that possible?
Select your earth, and add an empty at the same position. Hold shift, and select the earth. Enter edit mode. While editing the earth's mesh, add a vertex at the center of the earth. Press Ctrl+P and select make vertex parent. Now, the empty will have the same location of the earth, but will be able to rotate independently of the earth's rotation. Parent the moon to that empty. Now, you can animate the empty rotating to make the moon's orbit. Do the same thing with the sun. Add an empty at the sun's position, add a vertex in the center of the sun, create a vertex parent of that empty. Then, parent the earth to that empty. you can animate the empty on the sun to create the earth's orbit around the sun, and the moon's orbit will remain consistent.
An alternative to X-27's awsner is to use a circular Path: https://www.blender.org/manual/animation/object_path.html
Summary: Select the object that will follow the path, then Shift select the curve, and use Ctrl-P to bring up the parenting menu. Choose Follow Path. The object will now be animated along the path.