Many airlines charge you as much as $200 each way (delta), generally non-negotiable. Even I you have no other checked luggage... Go figure. Last I remember, JAL didnt charge anything.
Also, whether it is a small box where you have to remove both wheels or a big one where you remove neither, the charge is the same in my experience.
Otherwise: easy and cheap. Go to a bike store, ask for an old bike box. (alternatively, in a rush, you can tape together random boxes, though I don't recommend it)
Remove petals, front wheel, seat post and handle bars.
Deflate wheels. Put everything in the box.
Good to go.
If you want to be more careful, wrap frame in bubble wrap or plastic bags + clothing. Replace skewer on front fork to protect. zip tie or secure things in place. Google 'boxing a bike' for greater detail.
Also, travel insurance that covers flying with sporting equipment is not a bad idea. Some credit cards automatically give you insurance when you buy a ticket on them.
How carefully you box your bike depends on how much you care about the paint job and the material of the bike. I spend hours in my carbon Six but less than 30 min in my steel commuter (now ready for Tokyo streets and kusatsu).