Don't ever follow a dangerous animal into the woods because you didn't get its picture. You wouldn't dive into an alligator pool, and for the same reason, you shouldn't follow behind a cougar or a moose. And please don't pet a porcupine. Photography is like fishing in this regard. You often have to wait a long time in the right place to catch the best ones.
Patience applies equally when photographing pets and domestic animals.
They sometimes seem to have a knack for being uncooperative when you bring out your camera. Keeping calm and taking things one step at a time often helps the animal to relax. If you are planning your first trip into the wilds to take pictures of the furry, scaled or feathered residents there, it is a good idea to practice first. The zoo is an ideal place, because you will usually find the same kinds of creatures there. Study their behaviour and you will have an idea of what to expect when you confront them in a natural setting.