Schwinn's popular Varsity and Continental models featured the Huret Allvit rear derailleur. It's not a particularly durable or precise derailleur, and they commonly won't shift well after years of neglect. Never fear, I'll show you how to fix these.
First unbolt the cable from the pinch bolt on the derailleur. The cable threads through the bolt itself, and is sandwiched by this stupid metal clip that serves only to fray the cable and make it difficult to remove. Unscrew the nut holding the cable on there, then pry the metal clip apart with a flat screwdriver. This is what I'm talking about:
Once the cable is out (feel free to cut it if the cable doesn't slide out), remove the tension pulley using either a 9mm wrench or socket:
Now slide the chain off the remaining pulley and loosen the bolt on the derailleur's claw which holds it onto the bike's frame (if you have a chain tool, it may be easier to just break the chain rather than remove the tension pulley right away). The derailleur should now slide off the frame, and you can begin cleaning the parts:
The pulleys on these derailleurs are cool because they have little ball bearings in them. Unfortunately they tend to get gunked up and don't spin freely. They can be disassembled by putting two 13mm wrenches on the flat parts and unscrewing. Here's a closeup:
Taking the pulleys apart isn't necessary however, and the guide pulley (the one you didn't remove yet) doesn't come off without removing the rest of the derailleur cage. Instead I recommend just dripping some oil into the pulley bearings. The pulleys should spin easily afterwards. The rest of the derailleur can be completely disassembled, quite unlike modern derailleurs. The steel plates that move the cage back and forth are secured by nuts and bolts, unfortunately these tend to loosen up during use.
The barrel adjuster screw may seize in place on these derailleurs. If it is hard to turn, use pliers to unscrew it completely, then clean and grease the threads before reassembling. The spring can even be omitted if the adjuster is especially hard to turn. This screw is only used to take the slack out of the cable upon reassembly.
Here is the nearly finished derailleur in all its glory:
Make sure the pulley cage's spring is contacting the stop. If the cage just flops around, rotate it counterclockwise until the spring catches the stop. The derailleur is installed in reverse of its removal, by sliding the claw back into the frame dropout, removing the tension pulley, looping the chain around the guide pulley, then reinstalling the tension pulley with the chain in place. Again, you can also break the chain if you have the proper tool, and reconnect it once the derailleur is back in place.
Finally, thread a clean and oiled cable through the outer housing pieces and bolt it onto the derailleur. Check the limit screw adjustment with the back wheel on, and make any adjustments if necessary. The screw on the front of the derailleur is the high limit screw, and the long one on the bottom is the low limit.
With the derailleur reinstalled, it's a good idea to clean the plates that move the pulleys back and forth. I like to shift the chain into the biggest gear and use a
flat screwdriver to scrape the caked on grime from the derailleur's
moving parts. Lubricate each pivot point with oil or WD-40 so that the
mechanism can move freely: