1 key switch
3 brake lever
if your key switch is bad, then to test if it is the switch short the two wires coming out of the key switch and see if your scooter turns on. there is very minimal voltage traveling on these wires (4-5v) so if you strip it back and twist together with your fingers that would be fine, or if you have way to do it with out damaging the wiring great.
if you batteries are bad, then you might have lights on when the switch is turn/pushed but when you go to accelerate it immediately dies or shuts off, your batteries are probably dead and need to be replaced.
*caution* be very mindful of polarity when connecting your batteries back to your controller. these controllers are VERY sensitive to polarity changes so hooking them up backwards even for 1 second will most likely result in your scooter not working and you buying a new controller.
most all scooters use a brake lever for stopping. most of those have a second wire coming out of the lever going to the controller closing the circut and telling the controller to stop sending power to the motor. sometimes these switches go bad. remove the conector at the controller and see if that cures the problem. if not short out the controller side connector to see if that cures the problem. most controllers use a open circut to shut power down, but some use a closed circut with open being the trigger to shut power down.
controller, there really is no way to check the controller that i know of but there should be a slight/low muffled "click" when power is applied or the switch is turned on. if you can hear a click that means your is getting power. now check the voltage going to the motor when the throttle is applied, it should increase as the throttle is depressed/twisted. if it increases and then shuts off but resets shortly it could be your controller or a thermal breaker somewhere in the system.
Speed, Distance, Torque... Pick any 2