Planning Advice - regions- rental car vs public transportation

You do not want to have a car on the Amalfi Coast. You could drive down there and tec turn the car in Naples.

Watch a few videos of driving along the Amalfi coast and it will help you make up your mind. http://www.grandvoyageitaly.com/piazza/amalfi-coast-part-paradise-part-hell

Welcome!
Trains are easy for what you plan to do.
Way less stressful, much cheaper, etc than a car- which you don't need anywhere you are going.

It looks like you have 10 or 11 nights?
I think you'll enjoy your trip more if you cut out at least 1 location.
Pack light- easy for a short trip.

The Cinque Terre area, Tuscany

You don't have time for these places at all.

Rome needs bare minimum 3 nights.
Venice needs 3
Florence needs bare minimum 2 plus one for every day trip
Naples/AC 4- and sounds like you have that covered?

Every time you change location you lose at least a half a day. You don't have enough days to begin with to visit all these places. Minimize your travel time.

Venice and AC are both sort of outliers. You'll have to choose one.

Since you are flying RT to Rome- save Rome til end of trip. Eliminate that split stay/1 nighter
Fly to Rome
Head right to Florence- 3 nights
Train to Venice 3 nights
Train to Rome- take a picnic on board- 4 nights
Fly home

Or
Fly to Rome
Head right to Florence- 3 nights
Train to Naples/AC 4 nights
Train to Rome 3 nights

I'd say since you already have lodging in Naples area-- where?-- that's probably where you should go.

I'm afraid you're going to have little time for hidden gems, because you're just not spending much time in any one place. I'm not sure how you're counting your time. How many nights are you thinking of spending in Venice? You say you'll have "2 full days" there, but the Freccia trains between Rome and Venice (considerably faster than driving) take 4 hours, to which you must add the time to get to Termini Station in Rome and from Santa Lucia Station in Venice to your hotel. If you're thinking of two nights in Venice, that's going to give you just 1-1/2 days there. Even two real, full days (which would mean 3 nights rather than 2) in places like Rome, Venice and Florence isn't going to allow you to cover many of the cities' world-class sights, much less hidden gems. You don't see more my moving as fast as you are considering, you see less, because you spend too much time packing up, checking out of hotels, traveling to the next place, checking in to the new hotel and getting settled. As independent travelers, there will be no tour guide to hold your hands and get you efficiently from place to place, so it's not a good idea to travel at the pace of an organized tour.

How many total nights in Italy do you have available, not counting the night on the transatlantic flight?

In terms of transportation, the only place you've mentioned where a car might be really helpful is rural Tuscany. It makes no sense to try to drive in any of the other places you hope to go; you'd just be paying to park a useless car.

The way I am interpreting this is that "two full" days means three nights.
That is fine is you just want to see "the big three," but there is not time for Cinque Terre and Tuscany.
Car is of no help for the Amalfi Coast, but we need to know where you are staying and what you want to see to offer advice.

There's always the tension between the amount of time you have and the experiences you want to have. In an attempt to see it all you end up seeing nothing.

Take the 4 full days you mention and divide it between the big three you've mentioned. Moving from one place to another is a huge time suck with checking out, traveling and checking in.
With the 4 days you mentioned, add one to Venice, one to Rome and two to Florence, with a day trip from Florence into the Tuscan countryside.

Thank you to everyone who have sent their advice!
We made the decision to eliminate Venice and Florence (sadly) but we thought we were biting off more than we could chew. We booked hotels along the coast for 3 days (2 nights)- not sure exact town names right now but, Osodi, near Pompeii then making our way down to Amalfi wherein we are staying 6 days (5) nights and we are saving Rome for last 2 days since we are flying back home out of Rome.
Our last piece is the rental car which we haven’t completed as of yet.
We welcome all suggestions!
Thank you!

Nobody drives on the Amalfi Coast, take an high speed train from Rome to either Naples or Salerno and use the travelmar.it ferries as much as possible.

You have the timing a bit confused. There is no way that two nights give you three days anywhere. If you aren't traveling too far, it may give you about 1-1/2 days, assuming you don't wait until the evening to move on to your next stop. You need to be honest with yourself, of you'll over-plan for the time available, run yourself ragged and end up disappointed.

Again I really appreciate everyone’s advice. Here is the exact dates and locations:
Fly into Rome arrive 2pm staying in Lido do Ostia 6/28. - 6/30
Next stop Castellammare di Stabia 6/30-7/2
Next stop Vietri sul Mare 7/2 - 7/8 this location is approximately 2 miles from the coast we thought we’d like to rent scooters and tour the Spiaggia, Positana area,
Next we will head back to Rome 7/8-7/10 departing 4 pm
We originally thought we’d use trains but thought trying to catch trains not knowing how close we’d be to stations we thought it might be better to rent a car.
We tried to book rooms close to public transportation but worried even travel light it will be difficult with bags.

Driving a car anywhere in the AC is nuts
Driving a scooter is beyond nuts

Taxi from train station to your hotel.
Car rental is not a time saver in your scenario, quite the opposite.
If you want to go Vietri stay in Salerno, direct train from Rome. Vietri is so close to Salerno, I can walk there from Salerno.
Busses and ferry connections from Salerno to Amalfi.

mm285, It might be a little tedious work, but you can use Google maps to locate train stations in each town, and then have it show nearby hotels.

TripAdvisor has interactive maps that do the same. Pick a hotel, then click on the map, and you can expand to find the rail station and nearby hotels.

RS guidebooks also have small maps and list places to stay by location.

Driving a car anywhere in the AC is nuts Driving a scooter is beyond
nuts

Wise words, mm285. Summer traffic on the coast will be crazy for ANY sort of vehicle, 2-wheeled or 4. If you don't have considerable experience driving scooters, driving scooters in heavy traffic, or knowledge of local traffic signs /laws (scooter or car) please don't attempt it 'cause it's not for rookies. Plus, you'll be sharing the roads with these:

https://www.youtube.com/watch?v=2-S8coLeXcg

Finding places to park the car would also be a real pain in the (insert body part).