For photography, I would classify animals into three categories: (1) Wildlife (2) Domesticated (3) Pets.
I've read about, seen some television programs, and have some experience so I will tell what I know but will not necessarily have done it all firsthand. I will skip PETS because there have been excellent posts on this and would not be able to add to what's been said.
WILDLIFE: High end wildlife photography is relatively simple; spend $12,000 for a good 600mm lens and spend days and days and days hidden in some type of blind and wait for the animals to appear ("Patience" has been mentioned several times). The actual hard part is going to where the animals are. Want to photograph polar bears? You're taking a trip up north.
Most of us will need to be satisfied with the local species but you still need to go where they are located. If you have access to certain locations, you can set out bait to get the animals to come to you but I believe this is also illegal in some places. Deer, rabbits, squirrels, (the "common" stuff), if you're going to photograph those, they better be National Geographic quality to see any sales in the stock world. The database doesn't need another generic picture of a deer.
There are photography tours just to photograph polar bears and you can cruise to Antarctica to shoot (with your camera) penguins, but they are expensive. National Parks offer abundant wildlife right through the car window but again, it's a special trip that has to be planned for. There are a number of private parks where exotic animals roam freely but they are restricted access despite paying the entrance fee.
Suggestions for those on a budget:
If you live near areas with various wildlife and/or are taking a trip to such a location, search the internet for suggestions for where you are going. There may be tours available or just ask the locals when you arrive. I know places where the elk come down to graze during the evening at certain times of the year. Just pull over and hope they're close enough to get by without that 600mm lens. If anything, bring a pair of binoculars and enjoy the moment even if the shot you were hoping for doesn't happen.
Another trick is to place a bird feeder outside your window. Place it by a tree so you can photograph birds in a tree and not in a bird feeder. If you don't have a tree then find a large branch that you can mount one way or another in order to obtain a more natural setting.
Here's another fun trick for photographing birds: Set out a chair and put a scarecrow in the chair. Set one arm of the scarecrow on the arm of the chair and put a bowl of birdseed in the hand of the scarecrow. After the birds learn about the seed and become accustomed to the setting, remove the scarecrow and dress yourself the same as the dummy and sit down in the chair with the bowl of birdseed. You will have birds eating out of your hand! Hard to take pictures that way but it can be done. Fun thing to try for those with patience.
If there is anything I've learned about shooting wildlife, it's JUST HIT THE FREAKING SHUTTER RELEASE! Don't wait for the animal to move or turn it's head or wait until you move for a better background! Do what you can to improve the shot on the fly but TAKE A FREAKING PICTURE NO MATTER WHAT! Too many times the deer bounds off and I am left with nothing because the stupid deer didn't read my mind about turning it's head to the left just a little but more.
These are generally animals used in agriculture. I live in Wisconsin and many of you know this region is a large producer of dairy products, and dairy means cows. The funny thing is, when people think of Wisconsin, they think of happy cows grazing in a field with a quaint, old barn in the background. In reality, that doesn't exist any more. The dairy industry is going corporate and that means efficiency and that means money and money dictates the workings of the modern farm.
A modern farm consists of more or more giant pole barns which are ugly and the cows live inside, their feed delivered to them with industrial equipment. You can literally drive for miles through dairy country and never see a cow though thousands of them are right next to the road. You don't see them because they stay in the pole barn except when they're bored and decide to go outside.
I've been working on a project where I've been able to go onto the property and wander through these venues but they don't make for great photography. The light is bad, the setting isn't pleasing to the eye, and generic pictures of cows aren't very exciting.
I do have interesting shots from milking time but they are more "artistic" shots. That means noisy and grainy and we know what reviewers do when it comes to such factors.
Regardless, if you want to try a farm setting, one thing I've learned is all you have to do is ASK. Some farmers are suspicious, others think it's great. And you can get some good shots. I would suggest having a good flash with you.
Domesticated animals mean there are county fairs, horse pulls, competitions, shows, and what have you. A county fair is a fantastic place to shoot. Horses and cows are generally tied up where the photos are not very interesting but people will be interacting with them and that's where it gets fun. The participants have a lot of down time in between competitions and showings so there is a lot of activity going on.
When it comes to competing, prepping a cow makes for a fantastic shoot. You almost always have public access and you can photograph people preparing their cow. Hair spray, spray paint, even WD-40 is used to make a cow all shiny and spic and span for the judges. Now THAT is a picture, a person spraying a cow with WD-40! Or blow-drying it with a hair dryer!
I have many fantastic images from county fairs and horse pulls. They are not in my portfolio because they are meant for a specific project. Maybe I should glean through them again and throw out a few editorials.
One thing I've learned about county fairs is again, you have to be patient. I will walk in and out, in and out of the barns looking for something going on. You have to watch the people. You have to look for interesting light; the sun shining through doors and windows can do wonderful things at times. You need a schedule for the judging competitions.
Also, depending on where you live, keep your eye out for different events. I was driving down the road one day and happened upon a retriever dog competition. They let me go in despite not being member and shoot away. Other parts of the world will have horse riding events and local rodeos. Where I live there are special nostalgic events where you can watch people plow the fields with horses.
So don't forget about domesticated animals, they offer exciting possibilities for animal photography beyond wildlife and pets.
I hope this was of interest.
Nikon D800, D100, Canon G15