When it comes to photographing pets, if you're a DSLR owner all you really need is the camera body and a couple of lenses or if you're feeling creative, try using a fisheye lens for something a bit different. Not a DSLR owner? Well, your compact will fit the bill just as well, in fact if it's waterproof you'll be able to capture some interesting shots of your dog splashing through the sea at the beach (if you're brave enough to get your feet wet at this time of year that is!). Most compact cameras also feature a Pet mode now which is specifically engineered to make taking shots of pets easier. Plus, some compacts, such as the Pentax WG3, even have a Face Recognition function for pets.
The most important thing about photographing pets is to be patient. It's very rare you'll get the shot you wanted the first time you take a photo of them and it doesn't matter how well trained your pet is, as most are easily distracted and are often unpredictable so make sure you're prepared to spend a while taking plenty of shots. It helps if they're used to seeing you with a camera so they're not scared of it or worse still try and nibble it.
Make sure your flash is switched off as it could scare your pet as well as producing shadows that shouldn't be there. If you can't take them outside then find a room in your house that has plenty of natural light to stop your shots featuring unwanted, harsh shadows. It's also worth turning off any beeps your camera makes.
For shots with the 'ah' factor fill the frame with your pet's head/face, throwing the background into a nice blur as you do to make sure all the focus stays on them.
We're used to standing and looking down at our pets so taking a shot from this angle won't make it stand out from the crowd. Instead, try getting down to their eye level or even lower. For a more dramatic shot try laying on the ground and make use of LiveView to help you with the composition of the shot.
Before you take your shot, take a final look around the view finder to make sure you're not accidentally chopping off your cat's tail as this can spoil what would be an excellent image. As with any type or portrait work, you need to double check that the eyes are pin-sharp too.
Static shots are great but if you can get shots of your pet running and playing around they are far more interesting. Using continuous shooting mode can work well but if your camera struggles switch to manual and refocus on a particular spot and when your pet moves into it, press the shutter. You'll also need a reasonably fast shutter but not so fast that they're frozen. If you want to exaggerate their movement/speed try using a slower speed to add a bit of blur into the shot.