Oil based clay is the best clay for animation. The best brand is called Van Aken. Van Aken clays come in many colors, is very inexpensive, is animation proven and non toxic. Making new colors out of existing colors from this brand is extremely simple to do. All you will need is a double boiler, a large piece of plastic wrap, and a spatula.
First, boil some water in the double boiler, and place the base color clay you want to change the color of, in the top pot. Once it is fully melted, place different colors of small pieces of clay into the pot to modify the base color. Keep mixing until the clay is completely fluid, and adjust the colors accordingly. Next, place a long piece of plastic wrap on a level table top.
When you have the color clay you want, pour the still melted clay on the plastic wrap slowly, until it is all out of the pot. Any clay left in the pot should be quickly rubbed out with a paper towel so the next time you mix clay in your pot, the new colors wont mix with the old colors. One word of advice when doing this, is to make sure you have plenty of fresh air in your house because the oil can evaporate into the air and make you sick. Also, melted clay can stick to you if it is spilled on your skin, and will burn. Trust me, I have done it a few times and it hurts quite nicely. So use common sense, or ask an adult for help.
UPDATE 2013: Van Aken recently changed their formula and some have said it doesn't melt well. In the US, there is a new brand of clay that behaves similarly to the old formula of Van Aken. It's called Craft Smart clay, but we've confirmed it also does not melt into a liquid. It's manufactured in Thailand of all places, and has some properties that are better than Van Aken. From less cracking to less color transference from your fingers. In other words, the colors won't mix from one color to the next. You can get it in Michaels craft stores currently for a very affordable price. It also is more rubbery and great for fingers. However for faces we recommend using original Van Aken as the "rubberyness" won't flow out into parts of the face you don't want to move. For elbows and knees however it's great. Thank you Austin Williams and Tony Merrithew for this information!