Mistake 1: Carrying anything onto the plane
This should be a no-brainer, but I've seen parents with children also lugging their carry-ons on top of diaper bags and car seats and Ready for School Elmos. It makes no sense. Most likely, you already had to check a stroller, or some other equipment, and once you do that, you might as well rid yourself of all possible clutter.
The cheap way to do this, of course, is to go through security and then wait until the gate agents make that plea to check bags. Then magnanimously present your bags and check them for free. But think ahead. Pack whatever you'll need for the flight (magazine, iPad, tiny bottles of gin, etc.) amongst your child's stuff, or in your emergency bag (more on this below).