Take off the bottle cages, mounting bolts back into the frame.
Take off racks, mounting bolts back into the frame.
Loosen but do not take off bars and seat.
Turn the bike upside down, onto bars and seat.
Take off pedals, remember, one goes the wrong way.
Open the brakes and take off the wheels.
Take out the skewers and put them in your small parts bag with pedals.
Put blocks (free fom bike shops) in your forks.
Spread a piece of plastic and stack bike components as shown. The axles are slightly offset and go into the triangles.
Let a little air out of the tires.
Tie up or remove rear derailleur, removal is easy, remounting is a little trickier, ask for help if need be.
Reverse front fork.
Remove (loosened) seat and nestle into an open space.
Remove (loosened) bars and fit either over top tube or reversed front fork.
Bottle cages, pedals, tools, even shoes and helmet can all go into the empty spaces.
Tie everything solidly, several times, string is cheap.
Make a pad of a folded magazine or paperbacked book over the axle of the wheel to cushion it from your back.
Wrap the plastic up and around the bike into a package, use lots of brown packing tape. Make it strong.
Punch two holes, one under the top tube and one at the top of the rear triangle and thread through belts or straps.
Thread your bike gloves on the staps for padding
You now have your bike in an awkward but carryable backpack. You only have to carry it short distances to and from taxis and through the airport and train stations.
Slip the package into a bike box, either one from a bike shop or one at the airport.Tape it really well, bring your own tape to the airport. Leave the top open for security.
At the destination airport slip the package out of the box but leave it wrapped.
Carry it on the train as luggage and unwrap and assemble it at your starting point.
Pull the tape off the plastic, the plastic sheet will roll up quite small, and carry it with you on your adventure.
Buy more tape, buy more string and repeat the process for the return journey.

Many European trains now take bikes and a few American ones do but this way you can travel on trains, planes or buses anywhere, anytime with your bike regardless of regulations. The bike is also quite safe from damage. The forks are blocked and braced. the rims are set firmly against the frame with the front derailleur shielded. the rear derailleur is up out or the way. This is not difficult. Many, many bikers do this every year. Your bike will come apart like Tinkertoys and go back together the same way. Just be slow, observant and thoughtful.
NB Leftover bits are not good.

Or you could get a BIKE FRIDAY.