Taking photos of animals requires an extraordinary amount of patience and time. You can ask a dog to 'sit', but that doesn't necessarily mean he is going to look at the camera, if that's what you want him to do. Cats can either be difficult or easy, depending whether they are in their 16 hour sleep cycle or enjoying that fifteen minutes of terrorising the dog.
The trick is to have your camera either permanently attached to your face, or in an easily accessible place. I leave mine on the kitchen bench all the time, so if a photographic opportunity presents itself it's within grabbing reach.
Taking photos of animals when they are just hanging out doing their own thing is often best. You really get a true sense of their personality and capture them in a 'moment'. I used my zoom lens to get this photo of the cat. Having a shallower depth of field works well too.
I really like this shot of these horses. Try and think outside the square a little. This isn't your usual 'person on horse' type of photo, nor is it a standard sort of horse shot. These two were snuggling up together and it made for a different photo. I also cropped a lot out of this shot, their legs for example, which I think makes it work better. Think about what is within your frame. Sometimes closer is better.
It is, of course, great to get people in with your animal shots. Casual, un-posed shots work well.
Animals can create some awesome action shots!
And it is great to sneak up on them having a quiet moment. Remember that the same rules of photography apply, for example, taking photos of people during that 'golden hour' in the late afternoon will give you the best light. This old dog is highlighted beautifully in the early evening glow.
Zoom up on your subject. You will capture all those fine little details, like the feathers on this chook.
As well as the lighting think about what is in the background. These sheep were wedged in this pen waiting to be shorn. The old door in the background gives this photo a lovely rustic feel, and the light behind it really makes this photo. (so much so that I actually won 1st prize with this photo on the weekend!...very pleased!!)
Don't just stalk your pets! Get some shots while you're out and about. Cows make fantastic subjects! They're so mooo-ving. (sorry, couldn't help myself)
And don't be afraid to get dirty! Get down and in amongst it. You'll get some fantastic shots if you get down on an animal's eye level.