PaulRB:
How many colours do you wish to display? If you are happy with 7 colours: red, green, yellow, blue, cyan and white, then it is a little less difficult. If you want to mix more colours than that then things become more difficult.
If you wire up your cube as you describe in your breadboard, it will be very difficult indeed, because you will have 192 cathodes to control with digital outputs. If you want to mix more than 6 colours then these must be pwm outputs. I'm sure you realise Uno does not have this many outputs, digital or pwm.
If you multiplex the cube, as CrossRoads suggested, you will have 4 common anodes to control and 48 common cathodes. Uno does not have that many. So you need more led driver chips. 74hc595 are not really suitable, they cannot supply enough current. Tpic chips can, but series resistors will be needed. Other chips are designed to drive LEDs directly and do not need series resistors. Some of these chips have pwm outputs. These chips can be notoriously difficult for beginners to work with.
What I am trying to say is that this project is going to be about 50 times more difficult than you imagined. Building a 4x4x4 single colour cube is a difficult project for a beginner, RGB cubes are significantly more difficult.
If it were my project, I would consider using apa106 LEDs. These have controller chips built into each led, like ws2812b/"neopixel" LEDs, but are available as 5mm or 8mm package with leads.
Thank you for your quick responses and input. I'm thinking that your right and that an RGB LED cube might be a slightly bigger bite than my current skill level is capable of chewing, but I don't want to let that deter my efforts to learn how to do this because I think that this is a great learning opportunity and would be a good tool to have in my toolbox for later.
Ideally, I'd like to be able to display as many colors as possible. I also know that there are some compromises that i'm going to have to make strictly because of my current skill level.
The APA106 LEDs definitely seem to be the way to go for what i'm trying to accomplish. I didn't know that APA106 LED were a thing. So each individual APA106 LED has a built in shift register, am I understanding that correctly?
Makes me wonder why everyone else doesnt use them instead of wiring up a 2' wide wire ribbon to a bunch of shift registers. Am I correct in assuming that these APA106 LEDs would help reduce all of that noise and confusion?
So going off the assumption that these APA106 LEDs work the way that I think they work, Each LED would still need its own pin on the Arduino for them to be individually addressable wouldnt they?
As an additional note, I have ordered a INPUT EXTENSION SHIELD FOR ARDUINO to greatly expand my number of available inputs.
Once I can get this all sorted out and get a clear road map of what I need to do, I intend to create a guide on my entire process so that maybe it'll help the next person.