By my count, your group will be getting either on or off a train 12 times to go from the airport in Rome to Sorrento and then back north to Florence. It will take about 4 hours each way, not counting the time it takes to buy train tickets and find your way in the train stations and stop for food and bathroom breaks. For two luggage-carrying adults (or teenagers), not a big deal.
However, adding in jet-lag, a toddler, and the extra suitcases belonging to the three older adults, would make me re-think your itinerary. I see the temptation to rent a car and drive, but driving a minivan on that route immediately after your plane flight is even less of a good idea, for all the reasons mentioned already.
With only two weeks, and the group that you are dealing with, do consider visiting either Rome and Sorrento, or Rome and Tuscany. Not both. Yes, it is painful to cut out half of what you want to do, but I think you'll be happier overall.
I've never added it up, but I've probably spent a total of 30 weeks in Italy (much less than many people on this forum!) traveling independently all around the country by plane, train, bus, car, minivan, 9-passenger van, both with and without babies and kids, with and without being able to carry my own small suitcase, and in all kinds of weather and health. Just think through exactly what each small part of your proposed trip will be like and make sure you really, really want to take on so much. Just for two examples, will your toddler have a stroller that will also need to be packed onto the train and into a car? Will he or she have his or her own carseat or do you plan to rent one with the car?
Now, here's one practical tip that may or may not be obvious --- I can never tell. One way to handle the luggage on and off the trains (in addition to packing light and renting apartments with washing machines) is to make sure two of the suitcases are backpacks and that two of the others can fit securely onto two wheeled suitcases. That's how two people can handle six bags. This is how, these days, my husband manages to carry his own carry-on bag, my wheeled carry-on bag, his large messenger bag, my in-flight bag (my "purse"), and a bag with his medical device. The grandparents can be in charge of the toddler and watching the pile of everybody's bags as you get them on and off the trains.