Cheapest option near MIA often suggested on this board is the La Quinta East - nothing nearby except the Embassy Suites, but you can have dinner/pizza delivered. Good hunting!
Unless the mall is a big attraction choose the Mary Brickell area, with lots of local dining and shopping options. Taxi from MIA should cost about $25+ on the meter, or about $20+ using Uber or Lyft. The Hampton Inn offers a shared shuttle to the port of you can grab a taxi for about $15+.
You could stay near FLL and then take a shared shuttle like SAS to POM for about $15 each then next morning.
To get from FLL to your MIA area hotel you can take a taxi (about $75+) or an Uber or Lyft (about $55+, unless surge pricing makes it more expensive than the metered fare on the cab) or ride the Tri-Rail train (about $5 each) to MIA and then rely on the hotel shuttle.
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?