You 100% need a bag that will hold diapers. What that looks like varies depending on what you like and choose to carry.
I used a tote bag from Lands' End that I already had for the first four months, before deciding I wanted one that attached to the stroller handles (I walk all the time) and so I got my first Skip Hop bag (I love skip hop by the way)
I liked shopping for one at that point, because I had figured out what I carried and could try out bags by packing all my stuff inside. I know a lot of people who bought pre kid and it ended up being not the right bag for them.
In my bag I have spare diapers (cloth & disposable), wipes & wetbag. Wetbag with a change of clothes. Snacks for everyone and 1-3 waterbottles (I have two kids now). I have a zip bag of odds and ends, pacifiers, often a few small toys, my wallet, keys and cell phone. With a newborn I usually have a spit up cloth(s).
My husband has his own bag which he is in charge of keeping stocked after he kept stealing mine, with the wallet still in it.