My brakes on my car are squeaking when I press on the brake pedal, and the car stops quickly. Usually, when the brakes squeak, it means I need new brake pads, however, before I do that, I want to make sure this is the case for a car this old. Could it be something else? What can I check to see what is causing the squeak? Perhaps this is common for a car this old?

 

You need to Jack the Car up and inspect the Brake Pads. It is not hard to pull them out and take a good looke at the Old Pads and the Rotor.

If it is squeeking after you release the Brake Pedal there is a good chance one or more of your Brake Calipers is sticking.
Jack the Wheel you want to inspect up safely. Have someone step on the Brakes hard. Try to spin the Wheel. If it does not free up and turn easily Caliper could be Sticking.
If your Brake Pads are Extremely worn it is also possible for the Caliper to stick because the Piston is protruding too Far. This is more likely to happen if the Rotors had been previously turned down thinner during a previous Brake Job.

Loose Wheel Bearing end play can also cause noisy Brake Issues.

 

My rears squeak on the 240. I haven't fussed with them because they only do it when it's hot and not all the time.

I suspect they get glazed and that makes them squeal. I was going to pull the pads off and use 80 grit sand paper laid flat and rub the pad surface across, keeping the surface flat, to remove the glaze. Should fix the squeaking issue.