Travelling with Bicycles
My experience from the last years shows that it is no problem to transport trekking bikes by plane. Lower the saddle, remove the pedals, turn the handle bar by 90 degrees, decrease the air pressure in the wheels and wrap the bicycle in some protecting foil (about 5 sqm per bike).
Decreasing the air pressure is - technically speaking - not necessary, but airlines still enforce it (see German newspaper article).
Arriving at the destination, we have always thrown the foil away and bought some new before we flew back. In other countries you can buy it in do-it-yourself shops like in Germany. If nobody can advice you where to find a do-it-yourself shop, simply ask at the next tourist office. This has always worked for me.
You may also use bicycle cartonage from cycling shops, but then you can probably not use public transport to get to the airport...
Travelling with bicycles on long distance trains is a problem in many countries. Either it is completely forbidden (e. g. Sweden, France) or very much confined (e. g. Germany). The Deutsche Bahn AG offers a service to search for a connection that allows carriage of bicycles.
If a train is open to bicycles but does not have a carriages with bicycle stands for whatever reason, the conductor will find a solution for your bike. Though it may not be the best one as you can see on the picture.
Most countries allow two to four bikes at least on some local trains (but not in Sweden!). However you cannot book ahead which may cause an issue in popular cycling regions. I recommend to arrive early at the railway station to be there before all the others come. :-)