Unless there are fees involved and that's why you don't want to check more than one bag, I would check as much as possible. I hate having bags and stuff to keep track of on top of the kids and their carry on items so we prefer to check as much as we can while avoiding fees. We usually fly Delta and have the Delta AMEX which allows 1 bag per person free for anyone traveling on the same reservation as the cardholder.
As a PP mentioned, many people are trying to travel using only carry-ons now, so unless you board early the overhead bins are usually quite crowded. With Delta you are allowed a personal item and carry on for each ticketed person in your party, regardless if they are an adult or child. So if you want to travel that way, yes you can have a carry on for each traveler.