With a pencil and a page. Really—that's enough.
Coloring can be as simple as choosing the color that makes you happy and filling in a space, or it can become an art form involving layers, blending, shadows, highlights, texture, color theory, and dozens of other techniques. Neither approach is more legitimate than the other. The person happily coloring a purple chicken is doing it every bit as correctly as the person spending three hours creating photorealistic feathers.
If you're new to coloring, resist the temptation to learn everything at once. Pick a few colors you like, start with light pressure, and see what your pencils can do. Notice how the color changes when you add another layer. Try putting one color over another. Leave some areas lighter. Make others darker. Most of the techniques throughout the Art Studio are simply more deliberate versions of things you'll begin discovering naturally.
And yes, you're allowed to color outside the lines.
Choose a simple image with several reasonably large spaces. Before you begin, pick five colors—and put the rest of your pencils away.
Color the picture using only those five.
Try using each pencil at least two different strengths: once very lightly and once with more pressure. Layer two of your colors somewhere to see what new color they create. Leave one small area of the paper completely white for a highlight.
When you're finished, don't ask yourself, “Is it good?”
Instead ask:
What did my pencils do that I didn't expect?
That's a much more useful question.
Start lighter than you think you need to. You can almost always make colored pencil darker. Making it lighter again is considerably more troublesome.
Your paper matters. The slight texture of paper—called its tooth—holds the pigment from your pencil. Heavy pressure can flatten that texture, which is why building color gradually gives you more options.
You don't need every color in the box. Some of the most beautiful pages come from surprisingly small palettes. Learning to layer the colors you already have will take you much further than continually buying another 72 pencils. Although, admittedly, convincing ourselves we need more art supplies is one of the traditional hazards of the hobby.
Don't save the “good” coloring book until you're good enough.
You're good enough to color in it today.
Pages are meant to be colored. Pencils are meant to get shorter. And occasionally making an absolute mess of something is part of learning what works.