a. Is this correct?
b. Can I use the motor control board that comes with the car, or do I need to purchase an Arduino motor shield?
c. Do I need to worry about voltages/currents between the Arduino and the control board?
d. What kind of hardware do I need for all of this (breadboards, wires, resistors, connectors, etc.), and why?
a) we don't know, we don't know what you've bought
b) see a)
c) yes, very much so. Make sure they've got a common ground and don't exceed about 30mA per Arduino pin
d) don't know until you've answered a) and b)