Step 1: Catch Them Being Bad
Step 2: Find Out Where They Hide
Step 3: Catch Them at Their Finest
Step 4: Talk to Them
Step 5: Frame Them!
Step 6: Give Them Props!
Also, if you have a DSLR, a little bokeh is always nice. Try it with a quick DIY filter! Just remember, the shapes show up with a large aperture and with a single point light source - like string lights.
Props are awesome and easy to find, put them in a place you wouldn't usually find them...try a bucket or a plant!
Step 7: Kick It Old School
A good ol' fashioned "black and white" disposable camera from Walgreens and CVS come fairly cheap, sometimes the results can be surprising and it may result in a blue tone. Also they often don't charge you for the picture you aren't happy with when you're picking it up for processing - even if it's just because it's a bad shot, it's no questions asked. Just remember to look through the photos at the store!
If you are really daring try slide film. This is a very expensive option so shoot wisely! Processing it can be difficult as its hard to find a place to process it.
Film can be a bit pricey, so if you're going digital try throwing the white balance off on your camera to get a unique tone. Or you can make it black and white and hand color it with prismacolor markers!
Step 8: Be Silly, Give Them Clothes
Step 9: Get Closer
So try it! Photos don't always have to be of their cute little faces - catch their fur, a unique part of them like an extra toe if you have a polydactyl kitty, zoom in on their nose. There is so much more than your typical shot.
Step 10: Jump on in There
Step 11: Have Multiple Pets?
Step 12: Make a Montage
Step 13: A Couple Things to Keep in Mind
It may be tough, but try to get them in focus.
It doesn't have to look like it was done professional studio, it just has to portray your pet's personality.
Be creative and have fun!