My 2 cents, feel free to disregard.

If you have a Schwinn Varsity, that's a vintage bike, and half the appeal is supposed to be that classic look. A bike purist would think that painting the frame would be a sin, and replacing the parts takes it away from being "authentic."

I don't think a Varsity is going to make a particularly good commuter, mostly because of the poor braking (because of the steel rims). It's also heavy.

My advice would be to take your $300 and find a very nice used hybrid or road bike on Craigslist or a co-op. Get a knowledgeable friend to go with you, or ask on BikeForums to help you assess the value of a Craigslist ad. That money will get you a massive upgrade in the braking safety (IF everything is set up properly) and a huge upgrade in weight (10 pounds or more).

Then take the Schwinn Varsity and...
- If you need the room or money, resell it on Craigslist; you should be able to get just as much as you bought it for
- If you want a collector's item or the fun of having a project bike, keep it and take it to your local bike co-op and start the restoration process, learning about bike mechanics as you go