As someone above mentioned, check out the family message board. There are helpful tips and tricks mentioned by other parents/grandparents. One thing that is mentioned frequently is car seats and how to bring those on a ship. If you visit ports, how to transport kids with or w/o the seats. It seems to be an ongoing discussion.
The buffet will be a good option because they usually have some kid friendly foods available. All 3 of their cruises have been on Caribbean princess, which has world market place. On either side, they have a smaller cooler with milks and box cereal, which always was nice as a snack. This ship also has a new kids' area, which we haven't gotten to experience yet but it looks nice. We choose the ship over the ports because I found it nice to be familiar with the ship when traveling with kids.
The two main Princess pools are not suppose to be for kids using swim diapers. I've never used those pools with my kids but have seen others take their babies in them. From what I can tell, Princess staff doesn't say anything to the parents.
Bring more diapers and wipes than you think you'll need, just in case. You don't mention where you're cruising but in my experience, pharmacies are hard to find in Caribbean ports. Diapers and wipes are not sold in the gift shop.
Beach toys and other entrainment activities will need to be packed. You can't leave your child alone in the kids club until they are 3/potty trained. You can take them in and stay with them to play but it may only be certain hours.
Princess ships don't have changing tables in their public bathrooms so to change diapers, etc you'll need to take your child back up to your room. That can be difficult, considering where you are on the ship.
If possible, bring a grandparent or two to help you enjoy some time by yourself. It's not easy with kids and I wouldn't consider Princess the most kid friendly cruise line. My kids have cruised 3 times and twice I booked a suite to have more room for them to play/walk around in.
When you have the drill on the first day, my husband discovered that if you miss the first one, they have a second drill with less people and that was easier to attend with my son. I think he was napping during the first drill so we missed it.
Enjoy your cruise!!