How much does it cost to fly a balloon?

The answer can vary widely depending on what type of experiements are to take place, and how much equipment you already have or can be borrowed, but the short answer is about $1000.

Starting with the basics, you'll need a balloon, a parachute, flight packages, and some sort of tracking system. And it's hardly a balloon launch, if you don't come back with some stunning photographs of the edge of the earth, so you will definately want a camera.

Onboard the balloon is a tracking transmitter (with corresponding GPS position receiver), but you will also need some sort of ham radio receiver in order to pick up the APRS packets when the balloon is on the ground.  Many ham radio operators will already have equipment capable of receiving and decoding APRS packets, but if you have to start from scratch, this single item will likely be the most expensive part.