Here in the USA, it's easy to use Amtrak or Greyhound to transport a bike and rider. Here in the Northwest I know lots of easy out and back routes using the train or bus.
Flying with a bike is PITA, however. I can't think of a better bike to take to Europe than a break down Crosscheck. Heck, I can't think of a better do-it-all bike period! Take that Crosscheck with couplers, add trekking cranks, frount and rear racks, 4 small panniers and handle bar bag, 2 wheelsets (touring and lighter racing), hard travel case for bike...add a high end, ultra lightweight camping kit and you have one super sweet bike!
Yeah, I know a LHT would make a better loaded touring bike, and a road bike would be better for club rides and centuries, and a Soma ES would make a better light touring bike..... but that's 3 bikes! With the right set up, a single Crosscheck could do it all.
I'd set the total cost at around $3000-- for bike bike, case, 2 wheelsets, racks and bags, camping gear...and it would all fit a your closet.