I recently bought a women's Schwinn "Clear Creek" 18-speed mountain bike at a yard sale. The rear derailleur is broken, and the rear sprockets are rusty and don't turn easily against the wheel. I think I can replace the broken derailleur, but what should I do about the rear sprockets? I'm not so worried about the visible rust as I am about the fact the there is such resistance when I turn them freely against the wheel. Should I remove them and grease them somehow? Are there bearings in them that can be replaced?
Rusty sprockets aren't typically going to be an issue. As long as the teeth aren't too worn in, then the cogs themselves will be fine. However, if the cogset (likely a freewheel) itself is having difficulty rotating, it's likely the grease in the ratcheting mechanism has seized up, or the pawls are broken or faulty.
The internals of the freewheel might "wake up" if you try taking it for a spin, but it also might just be busted.
While servicing freewheels can be difficult, it is possible and the late Sheldon Brown could explain the process better than I could. The information is a little dated though, and might not be applicable to the part you have on hand.
Freewheels are a fairly inexpensive part, though there are some tools you'll need to remove the old one to put a new one on. If you don't have them and aren't willing to invest, your local bike shop would be able to mount it.
The busted derailleur is something you can probably handle, since that's usually just a couple of bolts to mount it. Watch out for the derailleur hanger though, since that's often damaged before, or with, the derailleur itself.
Something to keep in mind is that if your cogs are rusted, your cables might not be in great shape either. Look them over and pull on the brake levers. If they're unresponsive, stiff, or slow to retract, they'll likely need replacement. Gear cables will cause trouble with shifting.
Good luck and happy wrenching!