For domestic travel, you will not be asked to show ID for any travelers under 18. I fly with large groups of kids each year for theater trips, 4th-12th graders. None of them have ever been asked for ID. And, in all the many times I've flown with my own child (see sig for our MANY trips to Disney by air), I've not once been asked for any ID for him.
The only exception is kids under 2 flying without a seat. You may be asked to show a birthcertificate to prove your lap child is under 2 years old. Other than that, absolutely no ID required for the kids.