Wait. What?! As a Mechanical Engineering student, you should have no fear over something as simple as a bike. Look at the mechanics, and see how it Works. The shifter pulls a cable that is routed through a tube and pulls the derailleur one way. A spring pulls it back when you move the lever the other way. I give it a 90% chance that the whole works just needs lubrication. Start with WD-40 on all the joints. Consider buying a cable lubricating kit, which will let you force some oil down into the cable tube.
Consider this your first practical exam.
If you can't figure it out, bring it to the bike shop, and have them fix it while you watch.
If you're set on a new bike for a max of around $300, I just bought my Schwinn Coffee 3-spd for $330+ tax. I bet you can do better if you try. But it has everything a good commuter should have, except a high top speed. (plan on about 20 mph, which is actually a good clip on a bike) It has enough gears to handle moderate hills and flats. It has fenders and a spring rack, and a nice easy riding position. Best of all, the gearing is internal to the rear hub, so it doesn't get frozen by neglect as easily as an external derailleur system. Letting the cables rust in the tubes will still get you, but the rest is well-protected.
Anyway, I hope I shamed you into applying your engineering mind and fixing the bike. Even if you DO end up getting a new one eventually, you'll get more for the old one if it is in good working shape.