Best of Italy 17 Days to Amalfi

Great that you are adding on days.

You will need to be in your departure city the night before departure flight.

The RS tour is rather fast paced. All 2 night stays. That's fine on a tour but a bit more hassle on your own.
Do you really want to spend your extra 4-5? nights doing 1 night stays? That's a lot of time spent getting to and from, checking in/checking out, getting acclimated.

Stay in Rome the night the tour ends.

Pick a base- Sorrento is the best as far as access, transportation, etc.
From Sorrento you can day trip to Capri, Amalfi coast towns, Pompeii, Naples
Did you mean Paestum from Salerno?

Mondo Guides- highly recommended, affordable- for your day trips or can do on your own from Sorrento.
www.sharedtours.com

Rome
Day 1- travel to Sorrento- train to Naples then train or driver to Sorrento
Day 2- boat trip to Capri
Day 3- day trip Pompeii, Naples?
Day 4- day trip Amalfi coast towns- mondo guides, private driver? DIY bus/ferry
Day 5- Travel to your departure city

Day 6 fly home

I guess you'd need 5 extra nights. Unless you depart from Naples on a later in the day flight- Day 5.
Bus or driver from Sorrento to NAP

Day 4 We would spend the day in Salerno and Pompeii

Where do you plan to spend the night?
Pompeii is an easy 30 minute train ride form either Sorrento or Naples From Salerno its close to an hour

Are we better off having a base that we stay at each night(suggestions) instead of moving each day?

Two places at the most, otherwise too much of your time is spent checking out/checking in

I think Christine H's suggestions make a lot more sense than having to check in and out of a different hotel every night. Especially since all of these places are so easily done as day trips from Sorrento.

Day 1- travel to Sorrento- train to Naples then train or driver to Sorrento

You can also take a boat from Naples to Sorrento. The name of the private train company that operates between Naples and Sorrento is called the Circumvesuviana: https://www.visitpompeiivesuvius.com/en/naples-to-sorrento-train-schedule.

Throwing my vote in with Christine and CJean for this schedule:

Day 1- travel to Sorrento- train to Naples then train or driver to Sorrento
Day 2- boat trip to Capri
Day 3- day trip Pompeii, Naples?
Day 4- day trip Amalfi coast towns- mondo guides, private driver? DIY bus/ferry
Day 5- Travel to your departure city
Day 6 fly home

Moving to new accommodations nearly every day wastes time, and what you want to do on the coast is easily done from Sorrento; no daily hassles with luggage. I wouldn't do Capri from Rome, and yes, you want to be staying in your departure city the night before your flight.

I also like the idea of a Pompeii/Naples day, so that you can visit the archeological museum in Naples (closed on Tuesdays) where many of Pompeii and Herculaneum's treasures are housed. Pompeii may also be visited en route to Sorrento on Day 1, if desired. Luggage can be checked at the excavation or at the Circumvesuviana train station right in front of the main entrance. Naples might be just the ticket on a rainy day when you can do inside stuff like the museum, some churches, etc. Most of what you'd be doing on the Amalfi Coast is outside.

https://mannapoli.it/en/home-english/

2 and 1/2 days in Rome is really very little - my minimum recommendation is 3.5 just to scrape the surface - but maybe you'll be able to cover what you wish to see in that amount of time.

Yes to all the posts about staying in Sorrento all four nights. A word of caution about the train between Naples and Sorrento called the Circumvesuviana. It's dirty, crowded, and often filled with pickpockets. Especially if you have lots of luggage, you may be targeted. Consider taking the tourist friendly Campania Express instead, which runs on the same tracks but is cleaner and safer. Or consider transferring from the Naples Centrale train station to the airport via taxi or bus. Then take the direct airport bus to Sorrento, see https://www.curreriviaggi.it/en/naples-airport-shuttle. Another alternative is to hire a private driver to take you right to Sorrento from the train station. Also, I'd consider taking the Circumvesuviana on day trips, such as to Pompeii, without luggage as long as you wear a money belt.

You can also take a boat from Naples to Sorrento

While this is certainly true it adds 3 more "Seats" to the trip
Taxi from Napoli Centrale to port, ferry ride, taxi to hotel from Sorrento port.

(an option for riding ferry from Naples to Sorrento would be if you visit Naples while in Sorrento- take it both ways or one way)

Whereas the Circumvesuviana or Campania Express take you right to train station in center where you can likely walk to your hotel

Cheapest, more frequent
https://www.positano.com/en/naples-to-sorrento-train-schedule?timetable_route=sorrento_napoli

A few euros more, less frequent
https://www.eavsrl.it/web/campania-express-2021-en

Or if your budget allows hire a private driver who will meet you as you exit the train in Naples and drop you at your hotel door.
We used this one and were very pleased, about 100 euros (with 4 of you splitting cost that's not a bad deal.)
https://www.topexcursionsorrento.com/?portal_referer_id=46

The problem with the Campania Express is that it only has 4 runs per day so you could end up standing around for a long time before the next one comes. You might consider a private driver (as suggested in the post above) from Naples as it would indeed be cost effective for the 4 of you? I wouldn't personally take the ferry due to multiple transfers that involves.

If not taking it from Naples, you'll likely at least want to take the Circ. from Sorrento to Pompeii and back; thousands of tourists do that every year without incident, as we did. It's a battered, unlovely thing with no creature comforts but dirt cheap and gets the job done. Want to travel like locals? Climb aboard!

Yes, you'll want to observe any of the better methods of securing your valuables but that's a rule on just about any mode of public transit, in crowded areas or in-and-around tourist-attraction hot spots.

Love all of the feedback and suggestions. What would you see in Naples other than the Museum?

I gave Naples short shrift due to the nature of my trip, but now I'll have to return one day! So I always encourage people to read up on it to see what interests them--this is a great website: https://www.napoliunplugged.com/main
I'd personally like to see the underground and any Caravaggios in the city
https://www.napolisotterranea.org/en/naples-underground/
https://www.visitnaples.eu/en/neapolitanity/discover-naples/caravaggio-three-neapolitan-masterpieces

We had planned to visit Naples after Pompeii but the pouring rain that day got to be too much for us

I have this saved for whenever we do get there
https://community.ricksteves.com/travel-forum/italy/napoli-finally

You can also take a boat from Naples to Sorrento

The reason I mentioned it Christine is because I like Naples and the Circuvesuviana train when I'm not hauling luggage. Once you are beyond the gate at the ferry port you have nothing to worry about.