The starter gets powered by the solenoid which is energized when the ignition key is turned to the start position. Look at the picture below and read the sequence of operation below it. Once you find out what is NOT happening - you can troubleshoot back from there.
The IGNITION KEY SWITCH is connected by a small wire to the terminal in the GREEN circle on the SOLENOID. The battery is connected by a thick, heavy cable to the terminal in the RED circle of the SOLENOID. The terminal in the YELLOW circle of the SOLENOID is connected to a thick, heavy cable that goes directly into the STARTER MOTOR.
The -12 volts from the battery is connected to the car's engine, frame and body. The entire car has -12 volts available to all electric (and mechanical) parts, so only one wire is needed to energize those parts.
When the key is held in the START position, +12 volts from the battery is passed through the IGNITION SWITCH to the GREEN terminal on the SOLENOID. This causes the SOLENOID to (1) become energized, (2) internally physically connects the +12 volts RED terminal to the YELLOW terminal and (3) physically engages the STARTER to the engine's flywheel. When this happens, the +12 volts from the battery appears on the YELLOW terminal; and directly into the STARTER MOTOR. The STARTER MOTOR spins. Since it is engaging the flywheel, the flywheel turns too and the engine attempts to start. Once the engine is running, the key is released. The SOLENOID (1) becomes de-energized (2) opens the physical connection between the RED and YELLOW terminals, (3) disengages the STARTER MOTOR from the flywheel and the STARTER MOTOR stops spinning.
I hope this helps!