A hole in an interior door is a common site in many homes, and fortunately the fix can be done with basic tools and minimal time.
Remove any loose debris from in and around the hole. Depending on the size of the hole, you may or may not want to use spray expansion foam. If the hole is larger than a dime (1/2 in.), it is be best to fill the gap with foam. Follow the instructions on the bottle and spray the foam into the hole. The foam will expand 2 to 3 times its initial size, so a little goes a long way. Allow the foam to setup before proceeding to the next step (should be ready to trim in 30 minutes).
Using a putty knife (or razor blade), remove the excess foam, so that the foam is parallel to the door. Then, take your sandpaper and sand the foam into a slightly concave shape, which sits below the surface of the filled hole. Sand the area around the hole as well, to roughen up the area for good adhesion.
Using your putty knife spread the wood putty or spackle, over the area so that it covers about 1 inch beyond the hole. Allow the putty to cure, and then sand the entire area to a nice flat and smooth surface.
Using a spray primer (make sure to thoroughly shake the can first), spray a light primer base coat over the area. A white primer will be best suited for most door applications. After the primer has dried apply the matching door paint (a few coats may be required to fully cover the patch). Make sure you allow the paint to dry before applying any additional layers of paint.