Best advice I can think of would be the following two rules:
- Always focus on the animal's eye
- Mind your shutter speed
There are plenty more techniques when you want to get into wildlife photography, and/or pet photography...but those two will improve your shots a great deal.
Focusing on the eye works for closeups of animals, since the depth of field of many cameras can be shallow enough that not all of the animal will be in focus at once...so having the focus on the eye ensures that the primary part of the subject - the face - will be in sharpest focus. And it works for distant shots too - because often wildlife shooters have to use massive telephoto or telezoom lenses, and at those long distances, the depth of field at wider apertures can be very very narrow.
And for shutter speeds - always keep them higher - 1/250 or faster - if at all possible. Animals tend to make sudden, unexpected movements. Sometimes even small, subtle movements like twitching ears, or moving tail - or a quiver of the flank to remove a horsefly - results in a blurry photo because you weren't using a fast enough shutter speed. So by keeping shutter speeds fast, either in Shutter Priority or Manual modes, or by using a P&S camera's 'sport' or 'action' scene modes (in case you have a camera without manual controls), you can improve your hit rate.
Only special equipment I use is a really big lens! Wildlife is rarely sitting at your feet (and sometimes, you hope it isn't...if it can eat you!), so a big lens can pull in amazing closeups and details on some distant animals from a safe and unobtrusive shooting location. My main wildlife lens for my current camera is a 200-500 telezoom on a 1.5x crop camera body. So I've got 750mm of telephoto maximum distance, which is just enough for birding and wildlife.
National Geographic photogs and other professionals routinely shoot with 1000mm+ uber lenses that need their own tripods and are several feet long - and weigh as much as some of the big game they are shooting!
My galleries have lots of animal shots - I've got a wildlife gallery, some birding galleries, Everglades gallery, wetlands galleries, and of course tons of animals 'not quite in the wild' from my Animal Kingdom galleries. I've shot all the photos with an advanced P&S, or my entry-level DSLR - so don't feel like it can't be done without a professional setup. The big lens I use is still something hand-holdable, and under $1,000, so it isn't pro-grade stuff...but it does a great job for me, I have a lot of fun shooting, and even make a little money on the side from it.
His name is Stitch and he's one week old! We get to bring him home in 8 weeks!! He's the one on the left that has a similar resemblance to Mickey Mouse on his back!! And then when you're looking from the front he's the one on our left. 
I have a major weakness for German Shepherd puppies! He is just too cute!
So with that said I was just wondering if anyone has any expert or personal advice on animal photography? This can include your pet companions or animals in general (familiar or unusual) where ever they may be. 
for your imput and if you want to add some of your photographs as examples that would be great as well.
BUT these photos just make me happy!
