Car for 5 people

You could stay in Florence in a hotel that can accommodate a car (with parking), preferably outside the ZTL. I can suggest some.
No need to rent a car to see Siena, which is largely closed to traffic, you can go by bus on a day trip.
Same is true for Lucca, for the same reasons. You can go by bus or train.
Cinque Terre is best if you spend a couple of nights there. It's kind of too far from Florence (3 hrs by train, 2 by car) and you won't be able to enjoy it much on a day trip, especially if you want to hike the trails.
A car is useful to visit the Chianti and other smaller towns that are not served by frequent bus service or that require multiple buses (like Volterra). You could take care of those by renting a car for just a few days.
I checked on Kemwel.com for 3 random days in September, renting from Europcar. A full size VW Passat Station wagon or similar will be in the $250-$300 range, all inclusive (no deductible insurance). A car that size will be comfortable for 5 (I've had one) and actually has more cargo space that you can imagine. However you won't need cargo if you use it while you stay in Florence and do day trips from there. If that is the case, it's important to choose the hotel carefully, so that you can park at the hotel (or a nearby garage) for those 2 or 3 nights that you have a car. You could also stay in the city center while you don't need a car, then move to a hotel outside the ZTL (or in a ZTL safe zone) just for the period while you have a car.
A manual transmission is recommended if you want to keep the price low, also availability of automatics is more limited.

Great. So added to my previous post you also responded to (thank you) - I think I can fly into Venice (or Milan depending on frequent flyer mile limitations) and out of Rome and only need to rent a car for a few days and train/bus/boat the rest. I do think using Florence as a base will work, although I don't think I will necessarily go to much in Florence because of just too many sites/museums/churches for the teenagers. I need do a lot more research, but for right now I was just trying to get my start and end city, as well as number of days to stay on this trip since I have to get those tickets very soon. I may not get the exact number of days I want either, depending upon the availability. But with 3 kids, getting 3 tickets for free helps with the cost tremendously.

My thought is then:
Venice - 2-3 days and maybe side trip to Verona (not for Romeo and Juliet though - I know it's fake)
Florence/Siena/Chianti/Lucca 3-4 days (just need a car for Chianti and maybe a couple of other small towns)
Cinque Terre - 2-3 days
Rome - 3 days
Sorrento and surrounding 2-3 days
Fly out of Rome
So a minimum of 12 days and max 16. I was thinking 14 days for costs reasons. This seems to give me enough churches/museums etc while combining with some hiking and boating and hopefully some beach time as well.

Any other thoughts to get this down to the magic 14 (which I may not have a choice to do).

Count nights on the ground instead of days when planning.
Your itinerary then should be:
- Venice: 2 or 3 nights
- Florence/Tuscany: 3 to 5 nights
- Cinque Terre: 2 to 3 nights
- Rome: 3 to 4 nights
(total 10 to 15 nights on the ground)

If you add the gulf of Naples (Sorrento) then:
- Venice: 2 or 3 nights
- Florence/Tuscany: 3 or 4 nights
- Cinque Terre: 2 or 3 nights
- Sorrento/Amalfi Coast: 3 or 4 nights
- Rome: 3 or 4 nights
(total 13 to 18 nights on the ground)

The exact number of nights in each place is up to you.

Do the above in the order I gave. If Rome is the departure airport, leave Rome for last. A car advisable only for Tuscany (Chianti and small towns). Siena, Pisa and Lucca doable by train/bus. Pisa even doable as a short stop during transfer between Florence and Cinque Terre (by train).

I know this isn't your most recent post and am glad to see the plan coming together.