on GM fuel pumps, they push upwards of 65lbs of pressure, so the pump makes a high pitch squealing sound. this actually is normal, believe it or not.
the sound is from the pump spinning so fast it's oscillating/vibrating.
There is an aftermarket device made by Motormite/Dorman called the fuel pump pulsator. It goes on the rubber fuel hose and helps absorb some of the vibration and dumbs it down. Does it help? Yeah, it's only worth putting it on if you're installing a new pump though.
You can have your fuel pump pressure tested very easily with a fuel injection guage, you get the specs for fuel pressure for the Cobalt and make sure they're within spec. I'm sure it's fine.
Even replacing the pump may not cure the whine noise, it's honestly something that's normal.
To replace the fuel pump, you have to drop the tank, remove the pump, and just pop a new one in. Sounds easy.. it is, if you have a car lift.
from the ground it's very cumbersome. the cost for a new fuel pump is about $270 for an aftermarket one, an AC Delco is over $320.
There's no secondary fuel filter to worry about, and the $270 is a complete unit, not just the pump.