I got my Schwinn back together yesterday. All cleaned up and in once piece :-D.
I ordered the tubes and tires from Bikeman.com. The tires looked fine, but the tubes looked a little odd. Did you know they make different style valve stems? I didn't either. So, my new tires have the Presta instead of the standard Schrader. Other than that (which was solved by buying a Schwinn hand pump that works with both for \$10), I didn't have any problems.
I also sanded and repainted the handlebars to get rid of the rust, and sanded down the rims to do the same. Should be serviceable now (provided I get the tires inflated to the correct pressure now ;-).