+1 on the puddle pads. Nothing too expensive.

Cloth diapers. Even if you are using disposables get some cloth ones. They are great all around cloths for clean up. If you happen to have a boy, a cloth diaper strategically draped while changing a diaper can prevent a fountain (the kid looks small, but they hold an amazing amount!).

Don't go crazy overloading. Just the 2 infants are a lot to carry (particularly if you are on an outing as the solo parent). Say, after a few weeks, they are 10 lbs each. That 20 pounds of kid. We tended to strip down to the bare minimum for short outings - diapers, wipes, food & feeding supplies.

While the messenger bag may be good for looks, going with a day pack might be better for the dual shoulder straps - again, your hands are full of kids.

My wife and I developed great upper body strength toting two car seats around.