I am probably the only person in the world who misses her diaper bag. I miss it because I was ALWAYS prepared. In addition to the diapers and wipes it had other life necessities.
First of all, it didn't look like a diaper bag. It wasn't plastic with short handles. It didn't have cartoon characters or baby elephants on it. It was a Coach leather messenger bag in Black. Long strap, 3 separate compartments with zippers and snaps and two separate zipper sections on the inside. I got a screaming deal on it at the Coach outlet but I'm sure you can find some sort of similar messenger bag.
As I said I carried the mandatory diapers, wipes, small plastic bags for yucky diapers and bottle/sippy cup. If they are formula babies, measure the formula right into the bottle so all you have to do is add the appropriate amount of water and you don't have to carry around a formula can.
Also in my bag of tricks...a small first aid kit that included bandaids, eye drops, kleenex and anti-sting salve. A small swiss army knife is always attached to my keychain but if you don't, add that too.
Grown up and kid versions of Advil and/or tylenol.
Packets of peanuts, granola bars or gummy treats. Small containers of cheerios, animal crackers, raisins, etc. Doesn't matter what the snacks are as long as they can't melt or get squished. They save a crabby Mommy or kid when the blood sugar starts crashing.
Bottle of water for you and/or mixing the bottle.
A balloon weight so that the balloon you get at the restaurant does not fly to the ceiling making your child cry.
Take a M&M tube container and fill it with quarters, it's exactly the right size for quarters. You'll always have one handy when the kids want a treat or you need a cart at Aldi.
Safety suckers, these are great because they are completely sealed so no lint can get to them and they work for kids who need a treat and parents who need a hard candy because their throat is dry.