Arduinaught1:
Whenever I plug both leads of my dc motor into all combinations of the holes on the shield, nothing happens. Also, the leads are very loose in these holes, how am I suppose to do anything with the leads falling out?
Don't go poking things in random holes, the motors go into the screw terminal blocks, they should be labeled, the bare end is inserted into the opening then the top screw is tightened down to hold the wire securely.
That motor controller looks like the adafruit motor controller shield v1.2. You should be able to find all the info you need by searching for that instead of sainsmart. I'd be willing to bet it uses the same library.