Which order? Rome/Venice/Florence/Sorrento?

Is there any way you can change your tickets to open jaw (multi- city), and fly into Venice and out of Rome? It's usually in the same price range. If the airlines tries to change your flight times, etc., I'd push for a route change.
To decrease the checking in/ out of hotels, once you land in Rome, I'd just push on to Venice that day. We did that on our first trip to Italy, as we made the same mistake you did, and it wasn't awful. Just nap on the train.
Keep all your Rome days together.
Your question is asked weekly, sometimes daily- here on the FORUM. Do a Search here, like the attached example, review the many well-written Trip Reports, and read the detailed posts under Italy. You have a wealth of great info posted here to access.
You have obviously spent some time researching and your itinerary is well-paced. Train train is the best mode of travel.
Have a great trip and safe travels!

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Fly into Venice and out of Rome. I’d do Venice, Florence, train to Sorrento, then Rome.

Skip Sorrento since it is out of the way and spend your time to do a circle north of Rome and you'll get to visit the some of the many charming places in Tuscany and Ravenna, etc. without having to backtrack. Save Sorrento, which is more resort like, when you want to visit the south. You can book day tours out of Florence and Rome. You can take boat trips to other locations from Venice, like the Island of Milano. Check if there are speed trains available between the cities.

I am also flying round trip to/from Rome. Not my first choice since I need to train on to Venice when I land. Flying open jaw to Venice, out of Rome didn’t work with United’s scheduling. So flying RT isn’t a bad thing, just inconvenient.

I think you’re shortchanging Rome. Is this your first visit? I’d cut out Sorrento to add more days to Rome and one more to Venice. Your first day will be wasted with jetlag. Taking the train from Rome to Venice cuts over 1/2 a day.. You can add a day trip to Ostia Antica from Rome and a day trip to Padua from Venice to add variety.

As suggested above, if it is possible since you've already booked your flights, see if you can fly open jaw into Venice and out of Rome - or the reverse. Venice, Florence and Rome have so much to see and have many wonderful day trips from them that I would not not bother with Sorrento.

If your flights cannot be rearranged, I would suggest taking the train to Venice from Rome on the day of your arrival. Then work north to south. From Venice it’s easy to visit Padua and Vicenza, from Florence it’s a short train ride to Pisa and Lucca. From Rome some of the choices are Ostia Antica, Orvieto and Tivoli.

We are first timers to Italy in September, flying non-stop in and out of Rome with flights already booked.

IF the flights cannot be changed (as Pat suggested), then there are 2 options to consider:
Upon arrival in Rome proceed directly to either Venice or Sorrento and finish you trip with all the Rome days at the end.

Can you change your tickets to fly open jaw (multi-city) and fly into Venice and out of Rome?

Also check the price flying into Rome and out of Venice but make sure your flight home departs late morning not early morning so you don’t have to rely on an expensive water taxi from the lagoon to the airport. Instead, you can catch a bus from Piazzale Roma to the airport (30m).
What time does your flight arrive in Rome? The reason I’m asking is because there’s a direct train from Rome’s Fiumicino Aeroporto (airport) to Venice departing at 1:53p taking 4h 45m: https://www.trenitalia.com/en.html. Venice has two train stations and you want to get off at Venice S. Lucia (not Venice Mestre) when everyone gets off. If you arrive earlier (or later) then you need to change trains at the Rome Termini station (5h 15m). Sleep in Venice for three nights.
Take a direct train from Venice to Firenze (Florence) S.M. Novella station (2h 15m) and sleep in Florence for three nights. From Florence take a train to Salerno (4h) and sleep in Salerno for two nights.
On your full day on the Amalfi Coast, take a ferry from Salerno to Sorrento. When you’re ready to leave Sorrento take a bus back to Salerno but you may have to transfer in Amalfi. The bus stop is located in the train station’s cul-de-sac. You may have to wait until the second bus arrives so you can sit on the right facing the driver. These buses fill up and this way you’ll see the coast from the sea and from land. Wear motion sickness bracelets on the bus because it stops constantly on a curvy road, plus the elevation changes as you’re approaching Amalfi but the view is stunning. Sleep in Salerno for two nights. You can then take a direct train from Salerno to the Rome Termini station (2h 15m).

Your first day will be wasted with jetlag.

Sorry, I have to take issue with this statement - it would be nice if people spoke only of themselves and their personal experience. I arrived at FCO at 11 AM on 10/31 after a nonstop flight from LAX - my friends arrived at approx. the same time from Miami. We got to our apt near the Pantheon, waited for our host since we arrived early, settled into the apartment, then walked to the Pantheon, then down to Largo Argentina to see the cats. Then we walked down past the Victor Emmaneuel, taking our time along the Imperial Forums to the Colosseum, which we admired from the outside. Then bus back to the Pantheon, had a pleasant dinner at Miscellanea, stopped at a supermarket to pick up basic food items, then back to the apt. We were not wiped out at the end of the day.

When we first arrived we were stoked - we were wired - we had plenty of energy, and we're all over 70. So please don't automatically write off your arrival day. And it can be an ideal time for train travel, which takes virtually no effort from Rome.

We spent 10 nights in Rome, so I agree with others that you aren't allocating enough time there. Advice to start in Venice and skip Sorrento seems sound.

Your first day will be wasted with jet lag.----
Sorry, I have to take issue with this statement - it would be nice if people spoke only of themselves and their personal experience. I arrived at FCO at 11 AM on 10/31 after a nonstop flight from LAX

Mark, you omitted identifying where in the plane you were; it does make a difference.
It is true different people will have a different reaction. Probably fair to say most people getting off an airplane in Rome at what is 2AM to their body clock, are not going to be at peak performance.

joe32F -

I paid $120 for an extra legroom seat - the only upgrade available on the ITA A350 nonstop from LAX short of business class. No perks beyond a couple extra inches.

But my experience was the same arriving in Berlin from LHR on British Air - a connection from Air New Zealand, and after arriving in Bath on train from Paddington after Virgin Atlantic flight from LAX. So no extra comfy premium economy or business class advantage.

If there was an advantage, it was the A350 to Rome and the VA 787 to Heathrow - both are believed to help mitigate jet lag [but we know that jet lag isn't the issue on arrival day]. The Air NZ flight was on a 777, and then a cramped British Air flight to Berlin after 3 hours at LHR - still plenty of energy

I know that YMMV - I simply don't believe it's appropriate to tell others how they're going to feel after a long flight - or flights.

Agree with Pat that if you can change your flights to arrive in Venice and depart from Rome, you will cut down on wasted travel time. You will otherwise spend at least 2-3 days in transit from 1 area to the next. One could also cut out either Sorrento or Venice and add another area in proximity to the others. Italy is a smorgasbord and it's difficult to do everything in 1 trip.

If you keep your flights, I would do Rome to Florence to Venice, and then a full day of travel to Sorrento, keeping a final night in Rome.
Planning is the best part aside from being there. Best wishes...

Go to Florence upon landing (same day of arrival) by train (2 hours)
Florence 3 nights
To Venice by train (2 hours)
Venice 3 nights
Go all the way from Venice to Sorrento (via Naples) by train (7 hours). You could also fly from Venice to Naples, although you are not saving a lot of time once everything is factored in.
Sorrento 4 nights.
Go back up to Rome.
Rome 4 nights

Since you have already booked flights, the next step for me would be to put the actual dates for each city on a blank calendar sheet. If there are must sees that you have for each city, be sure that during your visit those must see attractions are open and take note of the hours. If not, tweak the dates of the base cities so you don't miss out.
Once you get that figured out, book the lodging. I booked my lodging two weeks ago, and had found that a number of possible selections I had bookmarked a week prior no longer had availability.

I’d cut a day from Florence and Sorrento and add them to Rome.

For places tonstay, you might consider staying in ‘convent/monasteries’. Originally for pilgrims, they are open to everyone. Good prices. My wife and I can’t imagine staying anywhere else. www.monasterystays.com is where we booked

My 2 trips to Italy (after living their 15 years ago) have been round trip to Rome.

First trip was into Rome...
Immediately to Salerno, 4 nights
then ALL the way up to Venice, 2 nights
Florence 4 nights
Rome 4 nights
Flying out of Rome
minimizing the wasted time of moving back to Rome the night before departure (using the one transfer to Rome to get there to stay and do stuff)

Second was similar, into Rome
Immediately to Salerno, 5 nights
ALL the way up to Lake Garda/Verona, 2 nights
Florence 4 nights
Rome 4 nights
Flying out of Rome

As I am always going to go to Salerno, and I know that I don't want to do anything I need to retain information or REALLY remember the first day and might be tired, spending 3 hours on a train isn't much of a problem to me. I have frequently wanted to punch people by 5pm on my arrival day when my excitement wears off.

If you're stuck with a round trip ticket, go far away immediately and then make your way back to your departure destination. No need in making your last day in Italy a transportation stress to get back to your original location, I would much rather enjoy a leisurely last day.