The 3x3x3-cube is undoubtedly one of the best introductory projects to learning Arduino stuff, electronics and programming. While one should have worked through some basic tutorials first to get the core ideas, one will learn a lot while setting up such a thingy. Or at least I did. The cube wiring is relying on shift registers, not because this is totally necessary for a 3x3x3 cube (which could be wired with an Arduino alone), but because it would in principle allow for bigger cubes, too. Please feel free to improve the sketch - if you have ideas for a better platine layout or so, I would be happy to hear about them. Just comment on the work ;)
For detailed instructions, please visit my blog (chrismicro.blogspot.de), you will find some instructions on necessary parts and a basic code example there.