If your Kenmore dryer is making a squeaking, squealing noise, the likeliest cause is that the belt is slipping. The thin rubber belt which encircles the drum that holds the clothes runs through pulleys and a motor, which turns the drum. Over time the belt can lose some of its elasticity and starting slipping against the drum surface, causing the noise. The belt may eventually need to be replaced, but a quicker, easier fix is to apply belt dressing, which is a sticky spray designed to stop belts from slipping.
Unplug the dryer. Remove the top panel of the body using your screwdriver to unscrew the upward-facing screws in the door opening. Remove the front section (with the door in it) by loosening the screws that are facing forward, which you can access after the top is off to give you unfettered access to the big metal drum inside.
Trace the path of the rubber belt encompassing the drum, following it down to the point where it runs through the motor that turns the drum. Lift the belt off the drum surface at that point as it runs through the pulleys to the motor. Put a chalk mark on the drum next to the belt, right where the belt lifts off the drum.
Hold your belt dressing can with the nozzle pointing at the underside of the belt where it's exposed with the lifting of the belt from the drum. With your free hand, turn the drum slowly while spraying the underside of the belt with belt dressing.
Continue turning the drum, slowly, while spraying the belt dressing onto the back of the belt. Look for the chalk mark that you made to come back around in front of you, indicating that you've rotated the drum one full circle.
Reassemble the dryer, putting on the panels and tightening the screws in reverse order from how you took it apart. Plug it in, turn it on and test it for the noise.