If you're going to spend money on parts, you will find that some make a big performance difference, and some are hardly noticable. On a tight budget, choosing your upgrades carefully is critical.
In your situation, wheels are probably the most cost effective upgrade. You will see the biggest performance difference, and they can be moved to another bike if you upgrade later. Get on ebay and look at the Rolf Vector Pro wheelset - I have seen them go for < $200 a set, and they're fast. You will eventually need a good set of high performance clincher tires, but save them for race day. They puncture easily, keep a set of average tires for training.
A good set of pedals and shoes. Clipless are the way to go - they take a bit of getting used to, but are comfortable and efficient, and can easly be moved to another bike later. I ride Campy Pro-Fit, but you might also check out Look, Time, or Speedplay. Check ebay for shoes - some times you can find a scuffed up but useable set of high end bike shoes in the $50 range, and you will need them for clipless pedals - they have the screw holes to mount the cleats. Sidi, Carnac are my favorites.
A decent saddle makes a huge difference, not so much in speed but in posterior longevity. You won't ride fast if your backside is sore. Ebay is a good source.
Cycling clothes - a couple of good jerseys and bib tights make a big difference. You're a lot more comfortable and dry, and after a couple of hours on the bike, that makes a big difference. Plain old shorts have a way of rubbing you raw in some pretty sensitive places. Ouch!
Cranksets, shifters, etc... if yours are working, then don't waste money replacing them. You don't see much performance improvement for the money you spend. Anything beyond wheelset, pedals, or saddle... save the money and put it towards a new bike.
Good luck - reminds me of when I was 18, had wheeled and dealed my way into a high end road bike, and was scratching for the money to keep a decent set of Clement cotton tubular tires on it. Yeah, that was quite a few years ago...