Animal photography and pet photography is actually one of the more fun styles. i know a lot of people that all they do these days is take photos of their pet. Let's talk a bit about how to go about it.
The first thing I like to keep in mind is that pets are like people. They're a member of our family, so we want to try and be at eye level with them if possible. We want to have that relationship with them, and we want to pass that relationship on to the person looking at the photograph. Another thing to keep in mind is that your pet doesn't get embarrassed like your wife. They're not going to be worried about how their hair looks or do they look fat in this coat. Animals have absolutely no sense of inhibition. You can take photos of your pet yawning, sleeping, stretching, eating, and you can definitely get some great photos from any different angle.
Other things to keep in mind, and this is one of my favorite activities when photographing pets, is to call them. Wait till they're feeling docile. Wait till they're lying down, and then call their name or give a whistle and watch their ears perk up, watch the head turn, almost like they're on the hunt. You get some remarkable photos of your animal in action mode as soon as you get to that. Most of these tips can apply to wildlife as well, with the exception of calling them and getting their attention.
If you're somewhere like the Serengeti or anywhere in Africa and an animal is there, the last thing you want to do is call it to come over to you. But maintaining that eye level, looking for them in their native habitat, having a bit of fun, and you will get great animal photographs