To keep my handbag/diaper bag tidy, I keep a large makeup bag inside. This makes it easy to keep all the kids' things together and keeps my bag mess-free.
It is also super handy when I go to the gym, I can leave the kids bag at the nursery and keep my bag with me.
Inside kids' bag I keep: 2 drinks, 2 diapers, 1 pair of underwear, Ziploc bag in case something messy happens, and snack box (this keeps snacks from getting crushed.. amazing!).