I went on a cruise with my family when I was 16...so I am reaching waaaaaay back here!
But,,prepare to be amazed! It really is a city on the sea. You won't believe how much room there is to wander, how much there is to do...
Whichever cruise line you are going with will likely have packing suggestions for the area you are traveling to. They also will let you know what is available on the ship. I didn't travel with little kids, but I almost wonder if they don't have strollers available for use on the ship? That would be something to ask about. There would be no problem taking a stroller along, but be prepared for a very small cabin! You spend little time there, so they really are quite tiny. So it may be ideal to take an umbrella stroller just so you don't have a big one taking up what little floor space you have.
We didn't need passports when we went to the Bahama's, so I wouldn't think you would at all.
Money..everything on the ship is included (typically), except tips if you feel compelled, and the duty free shops, gambling (our ship had a casino) and the bar. I am not sure if childcare is extra either...but going to the webpage of the cruise line you are going with will help.
You will want money for shopping on land, and any activities you want to do (we went to a really cool aquarium where you got to pet sharks and such, and a day trip to a fabulous dreamy place called the Garden of the Groves,,total tropical fantasy land)
Don't know about the cell phones,,,when I went, few people had them! We always just set a time to meet up at a designated place...
The daycare is your call...ask the ship how they find their childcare activity people. are they screened? Insured? Trained? You may be pleasently surprised to find that they are beyond qualified and a dream person to facilitate activities with your little one while you get an hour or so free.
I bet there are other mama's who have gotten to cruise more recently than I have.....

What fun you will have!