You can order supplies from Babies Travel Lite which will deliver them to the ship (they'll be in your cabin when you arrive) if you don't want to bring them with you. The cruise line does not supply gratis formula or baby food. There is a very limited supply for purchase in the gift shops. If you use a car service or rental car to get to the airport, it is easy to make a Wal-mart stop to pick up what you didn't want to pack rather than depending on it being available on board. Actually, unless you are on an airline that charges for bags, packing one case with formula and diapers is not a bad idea--you'll have it empty for goodies on the way home!
If there is a chance that you'll choose to do baby laundry on board, bring whatever detergent you use for your child. On board they will have All or Tide..no other choices.
As noted above, adult areas are adults only. Infants--sleeping, nursing, or otherwise are not permitted.