Know your airline dimensional and weight limits before packing. Print a copy just in case.

I split my coupled bike between a charity shop suitcase and cut down bike box for point to point travel. Full sized wheels are beastly to manage without a special case. The wheels go in the box.

My non coupled bike has only travelled by post and train.

Edit: Helmet goes on the head because I want to clue the X-ray people to what odd ball junk is in my carry-on. At one airport all the X-ray techs huddled together as the pointed at me and each object on the screen. I lol'd.