The rule is first get the surface clean and dull. So wash thoroughly, then scuff sand. The areas of dog chew you might be able to fill with Bondo, or the equivalent wood filler.

As for the details on how-to, watch this video, it pretty well covers what you want to know:

Be aware that a high gloss finish is going to expose imperfections more than a flatter finish.

I would sand all areas lightly, and then apply an oil based primer, such as Kilz. Apply liberally and let dry. Would be much easier to remove doors, then hardware and set up a couple of saw horses and make yourself a platform to paint on. Let the hinhes and handles soak in a bowl of hot water and liquid dish soap detergent over night and the old paint will almost fall off the next day. Apply two thin coats of paint, letting the first dry completely before applying second coat. On the areas where the dogs tried to eat the door, snd well and apply thin coats of plastic wood or wood filler. Sand this before priming entire door. Use a roller for the finish coat to get a nice smooth coat and let dry slowly.

Before you paint you must sand first, before you prime and paint. Use a 220 grit to scuff the cabinet door. Use a tack rag to remove and dust. Prime it sand lightly and put your final coat of paint on.