What would a basic setup for this be so I can learn and then possibly scale it up? Also and NPN transistor supplies current through it once I apply the voltage to the gate one it, correct? Does a MOSFET do the same?
For a 10x10x10 you'd need 300 anode drivers and 10 cathode drivers. Since only one cathode will be activated at a time the anode drivers only need to power one LED.
For the anode drivers you could use thirty-eight simple 8-bit shift registers like the popular 74HC595 and a current limiting resistor on each of the 300 output pins OR you could use specialized LED Driver shift registers that have built-in current limiting and possibly PWM brightness control.
The ten cathode drivers need to handle up to 300 LEDs each (roughly 6 Amps).
With an NPN transistor you connect the Collector to the cathode, the Emitter to ground and the Base through a current limiting resistor to a data pin. HIGH turns it ON. You need a transistor that can handle the voltage and current and which has enough gain that you don't have to draw too much current from the data pin.
With an N-Channel MOSFET the pins are calls Source, Drain and Gate. Be sure to get a Logic Level MOSFET that can be turned on fully with 5V or less. Typical power MOSFETs require 10V to turn on fully. You need a resistor between the data pin and Gate because the Gate acts as a capacitor and charging that capacitor might draw more current than is healthy.