travel order and other questions: Rome, Tuscany, Florence, Cinque Terre

If the Cinque Terre is your last stop and you want to fly home? Yes, try Milan (probably not Venice unless you really want to visit it for a few nights, and in that case I would start by flying to Venice and end in Rome). Also try Genoa and Pisa. Whether one combination departure city works better than another depends where you are flying home to, etc. So what works best for you may not work best for me. Sometimes it makes sense to end in another European city and stay the final night there after flying there from Pisa or Genoa or something (say if you can get a direct flight home the next day from the other city). Budget airlines may offer cheap flights from Genoa or Pisa next July but those flights probably for 2019 probably won't be on the schedule yet and not for a while.

I'd check to see what your options are and choose one based on convenience, price, etc. But it's probably too early to check airfares for next July at this point.

You should check AutoEurope to see what cities have available car rentals. I have found car rentals are easiest found in larger cities in Tuscany and Italy.

For our first trip to Italy Sept 2017, we had the same list of cities and 14 nights.. From Newark NJ, we flew direct into Rome. 3 nights Rome, train to Florence for 3 nights, train to Vernazza for 3 nights, train to Chiusi to pick up car for 3 nights Montepulciano, then drop car in Chiusi and train to Rome for 2 more nights. With the car, we went all over Tuscany and into Umbria. Returning it to Chiusi worked well since it is on the main train line. We would consider leaving from Milan if we re-ordered our cities, but leaving from Rome was not a big deal...

We have a one way car booked for Italy this fall, one week for only $154, including all added insurance (of course a "ForFour" car would be too small to fit 2 adults and 2 teens), but the point is, very inexpensive. Especially if you avoid airport pickup and dropoff.

That said, I like your plan, but why not rent the car from Rome (termini or other train station), then drive directly to Tuscany? It's a fun drive. We did it the opposite direction, specifying "no tolls no highways" on our google maps, and it was a great experience. It takes longer, and it definitely takes a navigator ( and you have 3!), but a great way to see the countryside and save on that train fare for 4 to Chiusi. A quick check on google maps shows 2.5 hours using highways (rome termini to Asciani), or 4 hours avoiding (meander and have lunch in a cool village on the way, since you probably have a 4 hour window anyway between departure and checkin at the agriturismo).

AutoEurope has a grid of all the car rental prices, and according to what I've read here and on Fodors, a good reputation for handling problems with billing etc. When you do the search, check carefully and they will show you all the available "rome" pickup points and prices.

Just a suggestion, I know it's not for everybody, but we loved those back roads in the Italian countryside. Elsewhere all trains and public tran.

So:
2 nights Rome (can you make it 3?), rent car drive to Tuscany
5 nights Asciano, then drive to Florence and return the car downtown and walk to your hotel.
3 nights Florence, then train to Vernazza
3 nights Vernazza, then train to Pisa
1 night Pisa (if necessary - if not, add it to Rome), fly home from Pisa.

Have fun planning! Use those know-it-all teenagers as your navigators!

Good advice above. I have one concern. Two nights in the capital city of Italy, with all its history and importance, gives you just one day to see things. That short changes Rome. Your arrival day could be compromised by jet lag as well. Take a night out of 5 Terre and add it to Rome. You will arrive sometime in 5 Terre, then spend an entire day seeing the other towns, and then depart. Two nights is all you need. It is charming but you run out of things to do quickly.

Another point, Venice is clear across the country from Milan, Genoa, Tuscany, and Rome. Go if you want, go, it is a magical place, but it is out of the way from your given locations.

The easiest way is to fly into Venice. It is a relaxing place to get over jet lag. Spend at least 2 nights, preferably 3.

You can take a train to Florence, spend your time there, rent a car to go to Asciano.
Drive to La Spezia, return your car there, and take the train to Cinque Terre.
Then take the train to Rome, spend your time there, and fly home from there. Easier to fly home from Rome, more flight options.
There is no extra charge to rent in one city and return in another, you are not crossing countries.

If you want to fly round trip to/from Rome, I think the following order would work well. Go directly to Florence upon arrival (2.0-2.5 hours by train), pick up a rental car when you leave Florence for your stay in Asciano, drop off the rental car in La Spezia and train to your Cinque Terre town, and finally take the train back to Rome for the last few nights. There are a few direct Freccia trains each day from La Spezia to Rome, and many more options with changes in Pisa and Florence.