Lodging for Rome & Sorrento - 3 Adults & 3 Kids (boys)

From your questions, it appears you haven't visited these areas before. I strongly recommend you fet a guidebook or 2 and read up. A car rental is to be avoided in both Rome and the AC. Rome has a metro system, and local buses. Taxis are of course available, although not all would accommodate 6 people. You might need 2.

The only train rides you would likely have would be the high speed freccia train from Rome to Naples, and the local Circumvesuviana train between Naples and Sorrento (the same train stops at the Pompeii scavi station for the ruins there. If the Circumvesuviana doesn't appeal, you could arrange a private car service between Naples and Sorrento.

To visit Capri, there is ferry service from Sorrento.

To visit the AC towns you can use the local bus service, which is usually slow, late, and standing room only (if there is even room to get on). Or ferry service + local bus. Or a private car service- but traffic on the AC is often bumper to bumper and very slow, with nowhere to park once you get to where you want to go.

and Uber around?

No. Italy does not have the US style uber.

You would need to use taxis or a car service, depending where you are.

Question? Where are you going after Sorrento? If flying home, from which airport? I'm asking as Sorrento or the Amalfi Coast aren't great places to be the day of an international flight from Rome. Most of us advise spending the night nearer the airport to increase the options available of getting to it, and decreasing the odds of a transport snag throwing a wrench in the machinery.

I'm also curious about Campobasso 'cause It's sort of an odd choice for first-timers to The Boot but maybe you've been to italy before? And is there a reason you've only given Rome 1 full day?

All that aside, no, you do not need and do not want a vehicle in Rome or Sorrento. It would be nothing but a frustrating, expensive headache.

You can take a train to Campobasso: search your options on the Trenitalia website. For the shortest journey you'll have 1 train change. From there to Napoli Centrale, look at a Trenitalia train to Napoli Centrale: there's one at 12:50 PM with just one change that can get you there in 3 hours/19 minutes. Consider a private driver from Naples to Sorrento. It's expensive but likely well worth it with 6 people to move, and to have station-to-door service. You'll need a van to accommodate 6 plus luggage. There is also a cheap, battered commuter train (Circumvesuviana) that it'll get you from Naples to Sorrento (train station-to-train station) with no air conditioning or other creature comforts. While some of us have been OK with it, it hasn't been exactly loved by others. There's also the Campania Express train that is supposed to be a step up from the Circ. and makes fewer stops but only runs 4 times a day, or ferry. The latter would involve transferring from Centrale to the port.

https://www.sorrentoinsider.com

As noted in one of your other posts, you'll use ferries and SITA buses to get around the Sorrentine/Amalfi Coasts, and potentially that bare-bones Circumvesuviana commuter train if intending to visit Pompeii. It's a perfectly fine choice for that particular journey.
(Edited for some typos and to add Naples>Sorrento ferry)