Mine are (a lot) older, but I still need to take stuff out and about for them (combination of disabilities and toddler)
I use a bog standard, rucksack. In it I have:
A nappy change kit - I use the Hip baby 'keep it together' kit from Babykind (designed for cloth nappies, I use it with disposables). I have had nappies/wipes/etc for 2 in here at one point - bigger children - so will be fine. I can fit 6+ large nappies in. How long are you going out for, that you would need more than that? If an all day trip, then you will need tomuse under buggy, but otherwise, a couple of changes for each of them should last you a good 4 hours?
Drinks - outside side net pockets, the ones for water bottles. Essential.
Medical kit - I have to have a variety of meds and food related stuff, all goes in front pocket of rucksack
Colouring books/crayons - to keep 2 of my 3 amused, and now also fit ipad for my eldest.
Packet of drinking straws - essential, in our family.
Various little toys - a couple of cars, a couple fo happy meal (shoot me now) toys, coloured dice, and an activity/sticker pad.
I have had, previously, changes of clothes, but I foind they were so rarely used that I gave up. Since I stopped carrying them, I have had to buy an emergency pair of trousers/t shirt 3 times. Over 6 years and 3 children. Worth it for the stress saved in cramming everything into an over full bag/weight when carrying it. I do have a full change in the car always, for each child (well, a ragbag assortment of clothes that can be made to suit), which has come in handy at times - usually due to car sickness!
But I found that even as babies, I rarely actually needed the spare set of clothes I carted about everywhere.
Snacks - the odd packet of rice cakes, or nakd bars. But then I've never been a big one for snacks anyway.