Since you will have time in between you could save a few $$ by using a shared shuttle (like SAS and others) from POM to FLL - should be about $15 per person. Then taxi or Uber from FLL to PE (about $15-20 for the cab). Your call.
Only hotels near the MIA airport (are you flying into MIA or FLL?) will offer airport shuttles. Those are a good option if you just want dinner and a bed, but you won't see much/anything of Miami. Hotels in the Blue Lagoon area near MIA are often suggested - Hampton Inn is a cross the street from a strip mall with dining, fast food plus grocery/liquor and drug stores for any cruise shopping you may need to do.
When do you cruise and what's your budget? Prices vary with the season - the $300+ room in Feb can go for $100 or less in August.
Water toys and beach at both will keep any kid entertained, but Paradise Beach is the more family friendly option IMHO, and also offers a HUGE pool, better facilites, etc.
It has been a few years, but we had no issues on the train; we ride during the daytime. Maybe a local can jump in with more current safety observations. As long as you can handle your own luggage and have time it's a good and cheap option.
You could Taxi to the Money Bar for good, but not great snorkeling from the shore - no admission fee, but you should buy some snacks/drinks. Then cab to Nachi for the rest of the day.
Taxi for about $75+, Uber or Lyft for a bit less unless surge pricing is in effect. You could ride the Tri-Rail train to FLL (call the hotel to find out the closest station) for under $5 each, then shuttle or taxi to your hotel.
Or call Larry's Limos and arrive in style!
Avis and Hertz have hotel locations within a couple of blocks of the port (depending upon which terminal), but a taxi to TPA from the port is only about $25 - why rent a car?
resortpass.com will link you to hotels/resorts offering day passes to their spa/pool/beach facilities. Tip them to store your bags while you are pampered, swim and enjoy lunch.
There are a few companies with downtown Miami locations, and you would need to specify which company to be sure. If a shuttle is not offered a taxi would only be about $10-12
There are a few, VERY FEW, Uber and Lyft cars with car seats, but you will pay more and wait longer. Your best bet is to return to the airport baggage claim area and use the cruise line shuttle bus to the port, which will be expensive and slow.