"...Beat-up looking Fuji...." careful there, Velocipedio, beneath that neglected exterior might beat the heart of a thoroughbred! Some of the older ones (pre-1990) can be some pretty great machines!
Hillary, sounds like you got plenty of advice! You know me, all I own is used stuff, so the idea of finding some used wheels would be right up my alley! I fixed up an old bike for the bride that has 27 inch wheels on it, and even managed to get new tires and tubes at one of the major discount super stores!
Good luck, and enjoy that Raliegh!!