We've sailed when our eldest daughter was 17 months (Wonder), then again when our youngest daughter was 5 months (Fantasy), again when she was 10 months (Magic) and most recently when she was 22 months (Dream). Hopefully she'll be 3 and potty trained for the next trip!
As previous posters mentioned, no swim diapers in the pools. This is a health code violation. Which ship will you be on? The splash pad on the Wonder is much smaller than the one on the Dream or Fantasy. I haven't seen the Magic's new splash pad, but I think it's been expanded.
Take an umbrella stroller. You want something small enough to navigate the hallways easily. Also, you'll be waiting around on elevators with a stroller and you'll want to be as small as possible since so many people cannot seem to walk a flight of stairs.
Something we learned over time was that you can leave your child onboard in the Nursery when in port, so if there's a shore excursion you want to do or just some shopping they'll be taken care of. We usually pick a really short excursion or shop for an hour or two if we get off in port. We actually have a harder time prying our 5 year old out of the clubs to go on an excursion than we do the 2 yr old.
Another tip is to book some nursery time during dinner time. You can find quick serve items to feed the baby before taking them to the nursery and then you can have a quiet adult dinner. We also tend to request a private table, since we don't like trying to carry on polite conversation with strangers while trying to get the kids to eat. We like to be able to concentrate on keeping down the mess and the choking hazards.
We also played nap time by ear. They were usually too busy and excited to want to sleep on schedule, but if it works for you its easier to transition back into real life later.
My husband also likes to get up early and push the stroller around the Promenade Deck for a few laps while letting me sleep in.
