How would you string these places together?

Your last city must be your departure city, in the example you gave, that would be Rome.
The order is:
3 Nights Venice
3 Nights Florence (1 day trip to Pisa/Lucca by train)
3 NIghts Cinque Terre (train)
4 Nights Tuscany Base (maybe San Gimignano?)-Rental Car
4 Nights Umbria Base (maybe Assisi or Orvieto?)-Rental Car
Return car in Rome or Orvieto (then train from Orvieto to Rome)
4 Nights Rome
Where would you rent the car from? Where would you return the car to?
Pick Up: LA SPEZIA (near Cinque Terre)
Return: ROMA (or Orvieto, but only Hertz is in Orvieto)
Where would you base for Tuscany?
Any place between Florence and Siena, but outside the two cities, since there are traffic restrictions in both. Near San Gimignano would be good.
Where would you base for Umbria?
I would stay in the Assisi/Spello area. But also Todi or Orvieto works. Hard to choose. It depends on which towns you want to see. Assisi is perfect for Assisi, Gubbio, Perugia, Spello, Bevagna, Spoleto, also Todi is not horribly far.
Orvieto is closer to Latium, but it would be good base for Todi, Orvieto, Civita.
I think Assisi or Spello is a better base to more places in Umbria.
Where would you prefer to fly into? Venice or Rome?
Better to fly into VCE and return from FCO.
Where would you prefer to fly out of? Venice or Rome?
FCO.
VCE (in the mainland) is hard to reach from the historical center of Venice (in the middle of the lagoon), especially if you have an early morn flight back, whereas from the center of Rome to FCO is a quick 30-35 min by taxi or train.

As usual, Roberto hit it right on the mark. Of course, he's from Florence. I agree completely.

Have a great trip.

I like the idea of moving a little further on your road journey every second day.
You save some time and some gas, and get to experience a new hotel.
It doesn't really matter but I would start in Venice and finish in Rome, which has more flight options for home.

Arrive in Venice - 3 Nights
Train to Florence - 3 Nights
Train to Cinque Terre - 3 Nights
Pick up rental car in La Spezia
Drive to Lucca - 2 nights (visit to Pisa)
Drive to Siena - 2 nights (visit to San Gimignano)
Drive to Cortona - 2 nights (day to Assisi+Gubbio)
Drive to Orvieto - 2 nights (day to Todi+Spoleto)
Return rental car in Orvieto
Train to Rome - 4 nights
Depart from Rome

Also, since you'll be going from Florence to Cinque Terre by train, you could visit Pisa on the route between train changes (you need to change trains in Pisa). 2 hours in Pisa are enough.

Lucca, IMO, is prettier than Pisa. If you want to take a day trip from Florence, visit Lucca only the whole day. Then you do Pisa on the way to the Cinque Terre.

I agree with Roberto's suggestions. I did visit Lucca and Pisa in one day, but it was a long one. I spent most of the day in Lucca and got to Pisa in the afternoon. Avoid the restaurants that encircle the Campo dei Miracoli, it's hard to find a good one there. If you decide to overnight in Lucca, try to stay inside the walls.

It must be good advise - the two guys from Florence agree!

I'm surprised that you are not going to Pompeii, since Archaeology was mentioned. (I skipped it too, when I was in Italy for 10 weeks, but that was because I was with a 4 year old.)
I would fly in to Venice, and home from Rome, particularly if you decide to venture south of Rome (I would add Amalfi Coast and Pompeii ad do less Tuscany/Umbria).
Do not even attempt to take a car near Rome or Florence!
Have a great trip.

Roberto as always has great suggestions. My only opinion from personal preference would be to swap CT and Florence. I would do Venice then CT then Florence. For me i'd rather breakup the big cities with smaller cities in between. After getting lost numerous times in Venice and all the history, go hike around the CT and explore the towns before going back into the Museums in Florence. Now the train trip from Venice to CT is a long day like 5-6 hours. If you don't mind that and leave early in the AM. Then you have Tuscany and Umbria before Rome. Just my personal preference.

Thanks so much for all the wonderful suggestions. We will fly into Venice out of Rome. I am doing some further research on Tuscany and Umbria to make better decisions about where we want to visit during that time. Then I can narrow it down to the best place to stay. It is really hard to not try to see southern Italy in this trip too. We would all love Pompeii and Herculaneum no doubt. I am trying really hard to slow down a bit and appreciate Italy at a little slower pace. We have done several other European trips and we tried to do too much. As Rick says...."assume you will return". LOL! We plan to see Amalfi, Paestum, Pompeii and Sicily on another trip in the future. Being school teachers we go on a long 3-4 week trip every summer. Southern Italy is on the bucket list! I will come back soon with more questions. I need to do some more reading.