It can get pretty addicting, even after considering you are not filling up your house with all kinds of animals! The first and foremost aspect I find that makes any photo of a pet look interesting is what the subject is doing. It will look pretty ordinary if your dog is found sleeping in a spectacular photo. But that same dog if caught floating in the garden by your camera could be a stunning shot. Snapping up puppies could yield some of the cutest photos. But to get really close to them first get friendly with them. Else they will really start to take defensive action in case you attack them. Close-ups of dogs and birds really look nice, and better still if they have an expression in their face. So bottom line comes to that if the pet is doing something in which it looks good, take photos at that time.
The other part is the background and the accessories around your pet. Often with cool looking and clean indoors there comes out some nice photos. Outdoors, a sunny day is a perfect time to take photos. Your shutter speed would also favor you. When your pets get playful in such an environment, there’s no better time and place to take your photos!