Italy Itinerary - Train, Rental Car, or Combination ?

I would do the following:
I like to have a car, therefore in your shoes I would probably fly to Venice and rent a car after Venice:
Arrive VCE
Visit Venice
Rent a car
Drive to Varenna, Monterosso (not Vernazza, since no parking in Vernazza), Siena area, Orvieto, Gulf of Naples/Amalfi Area, Rome.
Return car in Rome upon arrival. Fly out of Rome FCO.

If you want to stick to your plan, then you can do without car except for Tuscany/Umbria. In that case I would do:
Fly to Milan MXP or LIN> train to Varenna>Venice>Vernazza (all by train via Milan)
Or also fly to VCE and do Venice>Varenna>Vernazza (all by train via Milan).
After 5 Terre: Rent a car in La Spezia
Drive from La Spezia through Tuscany and Umbria (Orvieto) and all the way to Sorrento.
Once in Sorrento you can choose to keep your car (that would be my choice and probably I would stay on the Amalfi Coast with the car) or return it in Sorrento (in which case you should stay in a hotel in Sorrento and use Sorrento as the base for that area).
After you are done in that area, you can go to Rome. If you kept the car in the Sorrento/Amalfi Coast, you would drive to Rome and return it there upon arrival in Rome. If you decided to return the car upon arrival in Sorrento, then you would go to Rome by train (Circumvesuviana to Naples then Freccia to Rome).

sd,

"We're trying to make the best use of our time."

If that's the case, I'd suggest minimizing car time and using high speed trains as much as possible as that will be the most efficient use of your valuable holiday time. The high speed Freccia and Italo trains travel at up to 300 km/h, which is far better than a car.

All of the places you're visiting have good access via train or bus, while some of them are not especially "car friendly", such as Vernazza. Also, you may find that car travel now is different than 10 years ago. For example, most towns and cities now have ZTL (limited traffic) areas, and each pass through one of those will result in hefty fines! There are also automated speed cameras on the motorways, parking and high fuel costs, etc. If you're willing to commit to a specific departure on the high speed trains, you can save considerably be pre-purchasing tickets on the high speed trains, so that will save money over a car as well.

However, keep in mind that there are also some potentially expensive caveats to be aware of when using trains and other public transit in Italy. For example, Regionale tickets and those for Metro and buses MUST be validated before use, or you'll risk hefty fines. Also the high speed trains have compulsory seat reservations which are specific to train, date and departure time, and if you board the wrong train by mistake, again hefty fines.

If you are trying to make the best use of your time, I would also suggest flying into Venice, rather than Milan. I'm not sure how much time you have, but here is my suggested itinerary:
Fly into Venice, spend 2 or 3 nights
train to Varenna, 2 nights
train to Vernazza. 3 nights
train to La Spezia or Pisa, pick up car and drive to a Tuscany for 3 or 4 nights. To get the most out of your car, I'd stay in a smaller hill town near Siena, such as Montepulicaino or Cortona or an agriturismo in the surrounding countryside. Use this as your home base for exploring the area.
Drop the car in Orvieto and spend a night or two.
train to Sorrento for 3 or 4 nights
train to Rome, spend 3 nights
fly home

Use www.trenitalia.com for Italian train routes, schedules and ticket prices. There's no need to buy Italian train tickets ahead, but you can save some money on any of the high speed trains (such as Venice-Milan & Naples-Rome) if you buy ahead online. Just be ready to commit as most of the cheapest advance purchase tickets are non-refundable, non-exchangeable and are for a reserved seat on a specific train. Regional trains (such as Milan-Varenna, Varenna-Vernazza, Vernazza-La Spezia and Orvieto-Naples-Sorrento) can be purchased a day or two ahead and must be validated before boarding the train or you can get a stiff fine from the conductor.

It really depends on your desired mobility in Tuscany and Amalfi Coast as to if/where you choose to rent a car.

Arrive in Venice
Train to Varenna (Lake Como)
Train to Vernazza (Cinque Terre)
Pick up rental car at La Spezia for Siena & Tuscany
Return rental car at Orvieto
Train to Sorrento to visit the Amalfi Coast, Pompeii, and possibly Paestum
Train to Rome
Depart Rome

or

Arrive in Venice
Train to Varenna (Lake Como)
Train to Vernazza (Cinque Terre)
Train to Siena
Train to Orvieto
Train to Salerno and pick up rental car to visit the Amalfi Coast, Pompeii, and possibly Paestum
Return rental car en route to Rome
Depart Rome

You can also check train vs. car travel times by looking at Trenitalia plus viamichelin, or Rome2Rio. Keep in mind waiting times for connecting trains where necessary.

Don't be scared of trains. It's the most relaxing way to go. My biggest withdrawal problems after a European Vacation is having to be glued to my car again!

I did that exact same less pompeii and paestum in 2013 all via trains with no issues.

The trip from Venice to Vernazza was long around 6 hours. We didn't mind and left Venice real early 6:30 and got into Vernazza in time for lunch. If you leave early just make sure you check the vaparetto schedule in Venice to see what time the early boats run to get you to the station unless you walk.