Yeah I just cut and stripped a power cable. Be careful about which wire is which inside, check the color codes.
For the wires you mention, the short answer is "thick". The long answer depends on your application exactly, and how much power and current you plan to run through the cables.
I would say: minimum AWG14 for the motor wires, and AWG18 for the others. Personally I just put AWG12 for everything, makes it easy.