Watch the animal you want to draw. Look at pictures of both the real animal in the wild and other drawings of the animal.
Observe the basic shapes within the shape of the animal's body. Most often, these shapes will be oval-based, but from time to time an animal may have a more triangular body (cheetahs or greyhounds, for example) or incorporate other shapes.
Look for places where basic shapes might overlap in the form of the animal's body. Most animals won't be composed of a few basic shapes. Instead, an animal will consist of many shapes that overlap to form more complex shapes.
Draw the basic shapes in light pencil and then erase the lines where the shapes overlap to form a complete body.
Add details, including facial features, teeth, claws, paws, fur variegation, ears and tails. Just add the basic details here; don't start coloring in.
Give your animal cartoon-like features by making the eyes, mouth and nose exaggerated, as well as exaggerating other features like giving a dog large paws or ears.
Manipulate the eyes and mouth to give your animal facial expressions. The slant of the eyes upward or downward works for concern or anger, and making the eyes very large with small irises can show surprise. Experiment with eye and mouth combinations, and add details to the mouth like teeth and tongues depending on the expression.
Color your animal depending on whether you want the animal to look realistic or very cartoon-like. Even cartoon-like animals should simulate the color patterns of real animals to more effectively represent the animal. For example, if you want to make a blue lion, the lion should still have a lighter shade of blue on his belly and a darker shade for his mane and the tip of his tail.