Airlines differ in their rules. Some require boxing (you can purchase the box at the airport from Delta Airlines or some other airline counter for about $10). All require pedal reversal and turning the handlebars. A few require most of the air deflated. Always get there extra early since it may take a while to get hold of the box (if required) and to tape up the box with your bike in it. It's a good idea to call the airline and ask their rules before you go.
While some airlines charge extra for bicycles, others do not. Ask about bicycle policies and cost before you buy the ticket. Usually on international flights (except Mexico), you are allowed two luggages, and the bicycle counts as one. The other was my panniers tied together along with the sleeping bag.
On Turkish, Mexican and some other airlines, you just lower the seat, turn the pedals in, and turn the handlebar. No boxing! Some airlines (Turkish and Mexican) require most of the air deflated on the bike. There is no logic in deflating, but just do it since they say they must follow the rules.
You should always have the handlebar (and the cyclocomputer) slightly loose, to prevent damage during crashes or transport, and to make it easy to turn (without fidgeting for the allen wrench) for putting on trains or on top or bottom of buses when avoiding developed areas.
Update: a friend says he doesn't even box the bike; instead, he asks for a large plastic bag which he puts the bike in. He says the bike never got damaged.