"Never work with children or animals" is a maxim for a reason. Unfortunately, most pet owners (myself included) ignore this obvious logic and try and take photos of their pets. While you can ask a person to pose a certain way, your dogs are going to be far more interested in whatever food, toys, or squirrels are nearby.
With that said, let's look at how you can suffer through and take good pictures of your pets (or at least, try to).
What Makes a Good Pet Photo
A good pet photo captures the animal as they naturally are. You want something of their character to come through. If they're a bouncy, excitable dog, that's what you want to see in the pictures. If they're a lazy, regal cat who rules the house like their own personal fiefdom, then that's what you need to capture.
Good pet photos benefit from being natural. Staged, studio shoots can work, but I think they end up looking a bit silly. What dog lounges around in a perfect white room? My two would be more likely to have tracked muddy paw prints all over every surface. If you've got a dog or a horse or other outdoorsy pet, shoot it outside. If you've a house cat, rat, budgie or anything else, shoot it in its favorite lounging spot.
The Technical Stuff
Technically, there's nothing super challenging or specific about shooting pets. You can use any lens, camera or settings you want. I've used every bit of kit I own at one point or another.
One thing I've found works well is to use aperture priority mode with an aperture value of around f/5.6. This gives a shallow enough depth of field for the images to be interesting without having to worry too much about missing focus.
If you're shooting a relatively relaxed pet, you'll have a bit more freedom with your settings. For my more bouncy mutts, though, I like to keep the shutter speed around 1/100th of a second or faster. Just adjust your ISO until you get a shutter speed that's fast enough for what you're doing.
Other Tips and Tricks
Good pet photography is a volume game. The more pictures you take in different circumstances, the more likely you are to get a single, stunning shot. Whenever you're around your pets, keep an eye out for a nice image. Whatever gear you've got with you---whether it's a DSLR or a smartphone---use it to take the picture. I was resting out on a hike with my dog when I captured this with my iPhone.
By the same token, whenever you're shooting pets, you should use burst mode, if your phone or camera supports it. You'll get a lot more photos to choose from, so there's a better chance you'll have a great one.
One way to make your pet photos more interesting is to get down to their eye level. You'll see the world as they see it. If you just shoot while you're standing up, the only background you'll get is the floor.
Don't always try to take posed pet photos. Just shoot away while they play and do their thing and you might come away with something great. Below is the best shot from the burst above.
If you're shooting your pet outdoors, try and do it at a time with great light. The hours around sunrise and sunset work great. A cloudy day can also give you nice flat light to play with.
If you're shooting your pet indoors, try and position them near a window. Window light tends to create really flattering images.
If you've got a friend or family member around, get them to help you. They can help you "pose" the animals, or simply distract them so they'll look at the camera for long enough for you to take a picture.
Pet treats can work, but they're not always reliable. If I try to use them, my dogs just run up and get too close for me to take photos. See if they work with your pets.
You can always use Photoshop to remove any assistants or arms from your shots. The image below is actually a composite of three photos. One for the background and one for each of the dogs. My brother was holding them in place and I just edited him out!
Personality is the most important thing with pet photos. The image below was my family's Christmas card a couple of years ago. Sure, it's not the prettiest photo, but it tells you a lot about the dogs' personalities. A fun, imperfect image is better than a boring perfect one.
Last of all, relax. Your pets are never going to do exactly what you want. If you get stressed about capturing the perfect image, you won't enjoy the process or accept the results you get. For the Christmas card, I'd planned on capturing a traditional image but the dogs were having absolutely none of it.
Shooting photos of your pets can be hard work. At the best of times, they make bad models. At the worst...there's always the vet.
Still, every pet owner is going to try and photograph them at some point. You only need one or two great photos and you have years to get them.