My dog chewed the surface of my doors, how should I go about fixing this?
Hollow panel doors like the one in your picture can be purchased at your local home improvement store for relatively cheap, probably in the range of $30-50.  So the easiest thing to do is not fix them and just replace them.

If you are set on repairing it, you should sand down the damage, fill it with wood putty, sand again, the prime and paint (the entire door). With the texture and shape of the door, it is unlikely you could repair this in a manner that is not noticeable.  Just note that the cost of buying the necessary products to repair it are probably more expensive than just replacing it.