Unfasten the gear shifter and clean the inside of it with Q-tips. Spray WD-40 into the gear shifter to lubricate it. The grease inside the gear shifter becomes gummy and restricts motion over time. Reattach the gear shifter.
Remove any debris stuck in the Schwinn's derailer that might be preventing the gear shifter from changing gears.
Loosen the cable attached to the gear shifter if it won't advance to the next gear. Too much tension will prevent the gears from changing.
Check the derailer control cables for rust, fraying or damage to the cable housing. Have a bicycle repair professional replace the cables for you if the damage is extensive.
Examine the tightness of the screws on the derailer if the gear shifter keeps knocking the chain off the sprockets. Lift the rear wheel in the air. Switch to the gear that causes the chain to come off. Crank the pedals with your hand until the point right before the chain will come off. Tighten the top screw on the derailer and crank the pedals. If the chain still comes off, loosen the screw, place the chain back on the sprocket and tighten the screw even more.