I'll help you with the module.
The screw terminal side you'd connect power & gnd direct to your battery, via a fuse or breaker.
Then connect your motor to motor1 & 2.
The other side of the module with the 0.1" socket you'd connect +5v to the arduino's +5v pin, and gnd to the arduino gnd.
Then connect Dir1 to one of the arduino's digital i/o pins such as i/o 2, and connect Pwm to one of the arduino's Pwm pins.
Then use digitalwrite(dir pin) to control direction, and analogwrite(Pwmpin) to control speed.
Ground Dir2 & Pwm2, and leave motor2 outputs unconnected (unless you want 2 motors).
Using a relay if you only need start/stop is also possible, though possibly not much cheaper, work out the prices yourself.
Noise or voltage fluctuations might be an issue as Rons said, if your using a car battery and you use seperate power wires for the 'duino and motor driver module you might get away with it.