If you don't mind friction-mode shifting (which is all I ever use), you have lots of mix-and-match options and can easily perform piecemeal upgrades. Watch for clearance sales and eBay deals on components, do your own work, and upgrade opportunistically. Otherwise, you will find it more cost-effective to accumulate your lunch money towards a complete new machine.

Last year, I bought an aluminum-framed Ross mountain bike at a yard sale for $20. The hubs, pedals, and cranks were cr@p, but the frame was suprisingly decent. I bought several new components for it: Shimano/DT/Mavic wheelset ($85, Supergo.com), White/Sugino crankset ($50, sheldonbrown.com), Wellgo pedals w/ clips and staps ($30 at my LBS), SRAM PC-58 chain ($20, LBS), anatomic GT saddle ($16, eBay), and KoolStop brake pads ($16, LBS). I would put the final result up against any new bike in the same low-200s price class. (If I had patiently waited for the right deals on used equipment, I could have done the project for about $150.)