We have taken our 3 kids on 2 cruises this year. Our youngest was 17 mos on the first, 23 mos on the second. We had a blast on both. I won't repeat what everyone else has already suggested...I'll just make a few additional points.
Be sure to bring along a small bottle of dish soap for cleaning out sippy cups in your room. Also, if your child drinks lots of milk and you don't have a fridge (or don't have one that keeps it cold enough), get a bucket from the bar on day one. You'll want to get milk in the buffet, and may want to keep one on ice at times in your room, and the ice bucket provided is pretty small. When our steward saw that we were icing milk, he was always wonderful about making sure we had fresh ice for the bucket.
Bring some grocery bags for diapers. There are no plastic bags in the trash cans, so you'll want to be courteous to your steward (and, at times, your nose) by bagging up diapers in plastic for disposal.
On our first this year with the kids, the Fascination MDR had printed kids menus. When we sailed on the Paradise, we asked for them, and they said they didn't have any printed kids menus, but they were able to recite what was available. Do yourself a favor, and just print up a copy of the kids menu from the link provided earlier in this thread, and bring it along. If there isn't a printed copy available, you'll be that much ahead of the game. The waiter may not list everything when they recite it to you. (our favorite for the little one was the fruit plate for an appetizer). Kept her busy for quite a while. I have to say, we did wait an inordinate amount of time between courses on the Paradise - much longer than I was accustomed to on previous cruises with and without the kids. We ended up skipping dessert in the MDR, because our little one had reached the limit on patience, and simply grabbing dessert in the buffet.
On our first cruise with the kids, I didn't feel comfortable leaving my little one for the late night babysitting (just me being paranoid). My other kids love, love, love Camp Carnival. On this most recent one, we took her from 10pm - 1 am on 2 of the nights, and she did GREAT. We got some adult time, and she was none the worse for wear. She got to play with toys until she crashed - she had a ball. When you leave the little ones, they give you a ship-issued cell phone that you keep with you, so they can call you if there is any need for you to return to Camp Carnival (the only call we ever got was on the last night...and it was just a reminder to return the phone!!).
You'll have a great time!! Enjoy!!