Watch the Background
Think neutral -- a plain wall, not a cluttered cabinet, or a barn door, not a tool shed. Think contrast, but not too much -- a lighter background for a dark animal, darker for a light pet. If your dog loves to curl up on his paisley dog bed, consider throwing down a solid-colored blanket before you shoot. Your cat might look better against the solid green background of the lawn than in front of a busy garden bed. You might be able to edit a distracting background later, but it's easier to avoid it in the first place.
If you pet does something cute and you miss it, don't despair. Chances are, if you're patient and keep your camera ready, you'll catch an encore performance.
Children make the best photographer's assistants. Get a kid to help get your pet's attention with a toy or treat or by posing with your horse, dog, cat, or other pet. Nothing is more adorable than kids and pets together.
If you want to capture your dog kissing your child, do what the pros do: put a little butter on your child's cheek, and let the dog lick it off. Food is good for more than kissing -- it's also great for getting your animal's attention for the shot. Squeaky toys and laser pointers work well, too. If you can get your pet to stay still for a few seconds, throw a toy (or even your car keys) in the direction you want him or her to look.
Have Fun with the Software
The camera's just the first step to a great picture. Basic photo-editing software can do more than fix errors -- it can turn your images into art! Play with colors, contrast, sharpness, and more, or use special effects such as "watercolorizing" to create something unique.