My diaper bag always gets stocked up each evening so that it is ready to go for whatever the next day holds. In newborn life we really only go to the doctor, playground, grocery store, library and grandparents houses so I don't need to over pack too much. Also my daughter is no longer in diapers so that eliminates one item from my diaper bag. I like to use the Bumkin wet bags in my diaper bag to keep things separated, protected and easy to find.
Diaper bag - I personally like backpack style diaper bags so it's easy to be hands free.
Bumkin Wet bags - These bags are water proof are a nice large size to separate your kids items into to find everything easily. They come in cute patterns and make each bag easy to identify when in my diaper bag.
Spare clothes - I used a different bag to keep a full spare set of clothes for my daughter, a spare shirt for myself and two spare outfits for my son.
Snacks and water bottles - I use another bag to keep snacks and drinks for my daughter and I. I try to just pack simple items that don't need refrigeration for our small outings. I pack the thermos brand water bottle for my daughter as it is leakproof. My water bottle is usually just on the inside or outside pocket of my diaper bag to grab easily.
Wrap carrier - I love bringing my wrap style carrier for going on short trips out of the house and it takes up less room in a diaper bag then structured carriers.
Nursing cover - I usually have a nursing cover rolled up in the inter pocket of my diaper bag for feeding my newborn out in public.
Organization is key especially with two or more, I suggest attempting some sort of pouch method to keep your items from getting lost at the bottom of your bag or spare clothes from getting wet. If the wet bags aren't your style packing cubes, make up pouches or even large ziplock bags could work as well!