I have a now 3.5 yr old son who went on his second RCI cruise at 23 months. Im trying to remember if we even brought the stroller that time. If we did, we only used it in ports when shopping. He loved to walk/run around and it wore him out for naps later in the day (a must if you want to get through dinner). If you do bring one, bring a light weight small one or an umbrella type since you will have to fold it up every time you get in a cab, go through security on the ship and airport, etc. On the ship, we didnt use it at all and kept it in the closet. As for a car seat, we left ours in the car back home. Most cabs dont have seatbelts and sometimes I held on to our son for dear life! We took RCIs bus transfer to and from the airport and no seatbelts there either.
We did not use their babysitting service (there is a minimun age req too but I think yours is old enough), and with the kids club they have to be 3 and potty trained-no exceptions (we finally get to enjoy this on our holiday cruise). And unless your child is potty trained, he wont be allowed in the pools. Be sure to bring along lots of supplies (diapers, wipes, childrens meds, first aid supplies, sippy cups, dish soap, small toys-new work best, favorite blankie/animal for bedtime, sunscreen, hat, etc. We also bring along our own lifejacket for the beach (cant use the one on the ship) and some small cheap beach toys. We dont bother with excursions since they are too long, and most require a child to be 3-4 at least. Instead we shop in port, and head to the beach.
As for food and water...I was 2-3 mos preggo on our last cruise and didnt have any problems. If you go to Mexico or a similar country, dont drink the water, skip the ice (including Pina Coladas, etc), only get fruit that still has the peel on it, etc. The ship food/water supply is safe. I brought along a Brita sports bottle and filled it up every morning with water on the ship. Also be sure to wash your hands a lot (Purell just before picking up your fork to eat is good too). I got the flu while preggo and it was not fun at all. You should be over the morning sickness, but I found that I felt the motion of the ocean a lot more while preggo (I dont get sea sick, but maybe due to the extra fluid I felt it more than usual). It really hit hard once I got home taking me longer to recover from the sensation of still being on the ship. Be sure to get a copy of your records from your ob to bring along too just in case.
Feel free to email me if you have any other questions. Ive posted many times about things like this. Have a great time!