The easiest is probably with 4 planes of 16 LEDs, and 16 resistors. It uses a total of 20 pins (16 for the columns, and 4 for planes. You could do it without resistors, but you will be better off if you use them.
I started this thread about using shift registers with a 4x4x4 thread, you can find lots of stuff about running the same cube with 2 shift registers to control the columns.
Im also trying to build some RGB LED cubes, but right now I only have one cube built and working, and its a crazy "nothin but LED design", but it works, and has been running for months without breaking.https://sites.google.com/site/rgbledcubes/
There is good info there that you can adapt to any LED cube building, especially the stuff about wire and part prep.
There are a set of arduino tutorials for using shift registers, specifically the 74HC595, those are a good start. You can build the cube both ways, start without the shift registers, run the cube, then upgrade the cube to shift registers, and run the new software on it.
I did an instructable about upgrading 4x4x4 cube to using shfit registers at:http://www.instructables.com/id/4x4x4-LED-cube-upgrade-with-Shift-Registers-74hc5/
What resistor you use is dependant on what LED you use, typically a red LED wants a 150 ohm resistor, while a blue wants more like 100 ohms.