Another vote for the backpack. I found it easier to carry (and then run and chase after) DD with a backpack than a bag over my shoulders.
I just happened to have one like this, but I just looked it up and they are only $35 new:
Very durable and washable. I used bulky cloth diapers (cotton prefolds and homemade covers) and had enough room for one. LLBean also makes a size up if you need more room.
The other thing I did was sort of section off different supplies in smaller bags so it didn't turn into a jumbled mess. I like to just use ziplocs so I can see what's inside, and also so I have a backup for poop-splosions.