Well, if you want to actualyl print it like in the examples, you will have to provide exact info on what method will be used. Silk screen printing will require the artwork to be vectorized in Illustrator as may otehr printing techniques. Other than that there's a million ways to quantize and posterize colors, but depending on the source image, multiple adjustments and steps may be required to produce reasonably clean areas.
I don't quite have an idea of what I'm wanting to do this on yet, so for the sake of learning how what if I were starting with a similar image of a cartoon character? Perhaps something like this