I still love the messenger style bags for new parents, but I’ve now switched to a backpack style diaper bag. When your children are smaller, you’re often doing diaper changes in public bathroom changing stations. The messenger style bag is great to get things in and out of the bag while you’re still wearing it.
As your children get older, you’re carrying them more often and the diaper changes get less frequent. As such, I find that having both hands free to carry kids and having the bag squarely on my back works much better. I think the best backpack for dads with toddlers is the DadGear Backpack Diaper Bag. It is well constructed and has plenty of custom pockets and storage areas for all the gear our kids need. I love the visibility of the models with the reflective stripes.