Personally (5 kids, does NOT make me an expert) I always used a standard backpack. When they were little I'd carry them in a Baby Bjorn which is a front-carry baby sling, so a backpack is easy to throw on your back. When they were older and walking, same story.
I hated shoulder bags because you're always bending down to stop them from running into traffic (apple didn't roll that far from the tree...) and a backpack stops that. And there's plenty of room for diapers and the various extra bits, snacks, etc. When they get older this morphs into books to get them to fall asleep, a towel because MY offspring instantly beeline for any body of water like a freaking bird dog, no matter how muddy, hats, sunscreen, etc. Side water-bottle pockets and top pockets for quick access to bandaids (which I bought in bulk), etc. It's perfect.