I have often wanted to bring my own bicycle while traveling. Are there some international regulations regarding bringing my bicycle on a plane? What should I expect to pay?
Short answer: Yes, it's possible!
Long answer: You will have to inform yourself with the airline you're flying with if they allow bikes on the plane, how much it costs, and how big the box you put it in can maximum be. Also it might be interesting to know if they allow cardboard boxes (otherwise you will have to get yourself a hard-case box to put your bicycle in (which can be expensive!)).
A lot of good information can be found here:
Airline Baggage Regulation for Bikes (I'm not sure how up-to-date this one actually is, but it contains a list of a lot airlines and their regulations with respect to bicycles as luggage!)
Each airline has it's own rules. My friend carried his bike on 2 trips. Once it was free and the other time he had to pay some reasonable amount extra. He had to pack it/reassemble as compact as possible (turn and secure the wheel, flip pedals inward, etc.). He then packaged it in plastic tape (or whatever this thing people often wrap their luggage in is called) and dropped it at the oversize baggage counter.