Most US Domestic flights do not require children under 18 to have ID. However, if your children do not necessarily look like you then consider getting ID. I went to the DMV (drivers license department -- whatever yours is called) when DD was about 18 months old b/c we were going to Europe and I didn't want to have her passport on me at all times.
Also, DD looks like me and not so much like DH (mainly skin color) and luckily he travels with her alone but always has State issued ID and US Passport for identity purposes.
In general, it's a good idea to have some form of ID for your kids. There are places online where you can order a child ID, just search if you are interested. Of course, it's not a 'legal' ID but it's something.
Best wishes!