Most interior trim is made from softwoods, such as pine or fir, and it won't stand up for long to a dog that has mistaken it for a bone. You shouldn't set about to repair the damage until you've impressed your pet with the difference, but having done that, the procedure is straightforward. Even if the trim is badly gouged and cracked, the job isn't as difficult as it looks, especially if the trim is painted. If the trim is unpainted, and large portions of the wood are missing, you may want to consider replacing it.
Scrape over the damaged area lightly with a pull scraper to remove splintered wood. Avoid digging out splinters that are still attached to the wood and are repairable, but remove surface splinters that can be a problem when sanding.
Sand the damaged area by hand, using 120-grit sandpaper. Go with the grain of the wood and smooth out small gouges and nicks that are too small to fill. You'll probably remove some of the finish by sanding, which is not a problem.
Fill gouges in painted trim with drywall joint compound. This inexpensive, easy-to-sand material works as well as spackling compound or wood filler, and you probably already have some around the house. Trowel it in with a putty knife and smooth it as much as possible, then let it dry and sand it smooth with 120-grit paper.
Use epoxy wood filler to repair damage on stained or unpainted trim. Choose a product that matches the color of the trim or use a clear filler. Mix it with hardener according to the instructions that come with it and apply it with a putty knife. Let it set, and before it hardens completely, shape it with a knife. Sand it flat after it has completely hardened.
Apply wood primer to painted trim with a paintbrush, let it dry, then paint the trim. If the trim is unpainted, apply stain to the wood that you sanded, then topcoat with one or two coats of clear polyurethane varnish.