We will be arriving into Rome on the morning of June 25th. We will be
really tired that day! We will be there the 25th and 26th. We will be
traveling on the train on the 27th to Sorrento
So you only have 1 FULL sightseeing day (26th) in Rome and a jet-lagged part of another (25th) before departing for Sorrento. With that very little amount of time to work with, I'd do one of the self-guided walks in your guidebook, doing as much or as little as you can manage. I wouldn't do anything for that day which requires advance, timed-entry tickets, like the Colosseum or Vatican Museums, in case of a flight cancellation or delay + being too wiped to appreciate them. Fresh air will also help you stay awake.
I'd book an early-entrance tour of the Vatican Museums + St. Peter's for the 26th, have lunch, and then do the Colosseum in the afternoon: get timed-entry tickets for that one. All of the underground/panoramic coopculture tours are sold out for June but you can get ordinary-entrance tickets IF you try and land them when next they come up for sale:
RESERVATIONS OPENING
CALENDAR OF RESERVATIONS OPENING WITH GRADUAL RELEASE SCHEDULED ENTRANCE TIME COLOSSEUM TICKETS
DAILY OPENING
Every day new tickets for the next day at 3 p.m (italian time).
WEEKLY OPENING
Every Monday new tickets for the following week at 3p.m (italian time).
You might also buy tickets for the 45-minute tour in English: they are still available for the 26th. These may actually be easier to book than the tickets above and a good deal for the price. You can stay at the Colosseum as long as you wish when your tour is over.
https://ecm.coopculture.it/index.php?option=com_snapp&view=event&id=E6A1B5B3-BFDA-AC94-6929-0161B2E3A0C0&catalogid=BDB46B8C-0C49-10F2-6036-0167A8502B26&lang=en
Your tickets are also good for the Palatine and Forum so if you have any energy left (it is likely to be very warm end of June) you could do one of those. These two sites are, which are close to the colosseum, are connected to each other inside security barriers so essentially count as a single attraction.
With limited time, those early-entrance Vatican tours, while not cheap, are highly recommended so you can see the Sistine before it gets completely mobbed, be guided to some others of the highlights, and enjoy direct indoor access to the basilica. That saves a long-ish outside walk from museums to church, a potentially long, hot queue to stand in + another security check. A couple of the most frequently recommended tours:
https://www.walksofitaly.com/rome-tours/pristine-sistine-chapel-tour/
https://theromanguy.com/tours/italy/italy-best-sellers/Sistine-Chapel-Vatican-Tour
The journey to Sorrento will take 1 hour/10 minutes from Roma Termini to Napoli Centrale by fastest trains (no changes along the way), and another hour+ to Sorrento by commuter Circumvesuviana train. The Campania Express takes a few minutes less but only has 4 runs per day. Add time in Naples for switching trains + getting to your hotel and settled in once in Sorrento so you'll pretty much kill 1/2 a day.
Some folks like the ferry Naples> Sorrento but you would need to transfer from train station to docks, and they do recommend buying ferry tickets in advance during high season.
Info on the Circumvesuviana, Express + ferries here:
https://www.sorrentoinsider.com/en/getting-to-sorrento-italy
And lastly, some folks use private drivers from Naples> Sorrento. This is the most expensive option but may be worthwhile, cost-wise, with 4 people. It may very well not save any TIME over the train, though, if traffic is very dense...as it is known to be in high season.
(edited to suggest the self-guided walking tour)