Photographing wild animals and pets can be a chore but with these few tips it will make for a better experience and better photos!

Items Needed:

Camera (film, digital or phone)
pets or animals
patience

Step 1: Tip 1: the Camera

Film/Digital Camera:

Always have the Red Eye Removal turned on (if you have that option) to avoid demon looking animal eyes. There will be times this isn't enough, if that is the case try taking a picture of the animal looking off to the side or at something if possible.
If possible take the photo outside, natural light is the best lighting and this also helps to reduce the chance of having demon eyes
When using flash animals tend to blink at the blinding light so again outside light is best to avoid the flash but if you cannot avoid it then again having the animals attention on something other than you and the camera will hopefully keep them from closing their eyes and avoiding the red glowing demon eyes
Tripods are nice for stability but animals move and you need to move as well so a small light weight tripod is best, or you can go without one and just trust your steady hands.

Phones:

Same applies to using phones for your camera, if you have a red eye removal option use it
Only use flash when necessary
Light and small tripods work best

EXTRA: If you follow these steps and you still get red eyes, try using a photoshop program to fix them!

Step 2: Tip 2: Photographing the Animal

Pets:

Take the photo from their level (down low) *Caution: if you are down low and face to face with the animal this may be misconstrued as an aggressive move, so know your animal well*
Don't be afraid to get up close to the animal (see above caution) or if not possible let zoom be your best friend!
When you need to zoom in a tripod is helpful in keeping the camera/phone steady to avoid blurry and out of focus pictures

Wildlife:

Zoom is your friend - wild animals are tricky to photograph because they tend to spook and flee, so using your zoom to get "close" is the best option it also keeps the predator animals at a safe distance
Patience is a virtue (applies to both pets & wildlife)
Stealth try to be as quiet as possible to avoid spooking the animal (if you have a noisy phone put it on silent)
Sometimes you get the animals that are not afraid of people use caution when photographing these animals (they are still wild and can carry disease & can attack) always stay standing, if you want a low shot crouch down a bit and hold the camera lower or use a short tripod and don't make any sudden movements, go slow!

EXTRA: If you are taking pictures of an animal behind glass like at a zoo don't use the flash it reflects off the glass and ruins the picture, take at an angle if possible to avoid bad reflections

Step 3: Tip 3: Bonus & Extras

Be mindful of whats behind the animal being photographed, and ugly wall or strange person can ruin the whole photo!
to add "drama" you can make the photos black & white or sepia toned if your camera does not have these functions a paintshop program can do this as well
if you do end up with a strange thing in the background then if you can successfully crop it out then do so!
Using the sport or action mode on your camera helps when photographing a animal in motion (running jumping or flying) no blur
Don't be afraid to take a bunch of pictures instead of waiting for that one perfect shot
Be mobile, animals move around so should you
The animal does not need to be front and center in the picture off center can make for a nice "artistic" shot
Don't worry about getting the entire animal in frame, head shots are wonderful too especially if you capture a wonderful expression

*That's all from me! Hope you like my tips! HAPPY PHOTOGRAPHING*