We do elimination communication, so our daughter only wears diapers when we're out or when she's sleeping. We use regular cotton flats.
For a newborn, if you're a good mommie, and change whenever baby relieves himself, you'll probably change every half hour when he's awake, to an hour. So you're looking at needing probably 60 flats or so, if you double-up.
We just do them with the rest of the laundry. They clean very well.