You have two options. The first is to cut from different colors of vinyl then apply them over each other like you see below. These were cut from different colored vinyl then over laid using transfer paper.
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The other option is "print then cut". This is a function where you buy white printable vinyl and use the Cricut software to print a multi colored image. Once printed you can place that in the Cricut and it will find the edges and cut the design. I have the printer and the printable vinyl but I haven't spent the time to mess around with it as the above method has worked for everything I want to do so far. I will need to figure it out soon as I have a FoxFire Upscale planned and those are too complex to piece together.