I normally try to stick to ground transport - normally trains, or occasionally buses. Of course, using the train is reasonably straightforward within Europe (except that every country has their own bike regulations, which differ by the type of train and sometimes time of day/week). However, I've also done this within North America pretty frequently, using Greyhound in the US and VIA Rail in Canada.

In Europe, the bike either stays fully assembled, or I have a simple transport bag that weighs only 1kg that the bike goes in with only the front wheel removed, and then goes in the luggage area of the carriage as a standard piece of luggage (particularly useful on the high speed trains, which don't always accept fully assembled bikes, but always accept them in one of these bags) - the bag folds down really small afterwards. In N. America, they frequently only take bikes in boxes. Getting the bike and a box from the bike shop where I picked up the free box to the bus/train station is a real pain. The box then gets dumped once leaving public transport (ask the guys at the station where their cardboard recycling is).