Using a GPS in Itlay

VENICE TO CUNEO: It's at least 5 hours drive. If you don't need a car in Cuneo, you might consider taking a train (via Turin). There are some solutions that require only one train change (in Turin=Torino) and you can be in Cuneo in just over 6 hours without stressing on the freeway for 5 hours. If you need a car in Cuneo, you might also consider a train from Venice to Turin, then renting a car in Turin near the station. It's up to you, just consider costs (if several of you traveling together, the car might be cheaper), time and your willingness to drive a long distance on the freeway.
CUNEO TO MONTEROSSO: For sure you won't need a car in Monterosso, but if you have a car already in Cuneo, you might as well keep it and drive to Monterosso. The drive is over 3 hours. If you don't have a car and want to take the train the trip is about 6 hours (back via Torino and Genoa). Some solutions require only one train change (in Turin).
MONTEROSSO TO REGGELLO: You will need a car in Reggello almost certainly to go around Tuscany. If you already have a car in Monterosso (see above), then drive to Reggello. If you haven't rented one so far, in your shoes I would rent one in La Spezia, which is just 20 min. by train from Monterosso. Then I would drive from La Spezia to Reggello. You could also take a train to Florence and rent the car in Florence, but from Monterosso to Florence is a pain on the train, because you need to change trains at least twice, in addition driving in Florence is a thing that most people like to avoid altogether (and that includes many people I know who live just outside of it).
REGGELLO TO SORRENTO: If you followed my advice, you will have a car in Reggello. I would drive it to Sorrento because the train option is too convoluted. Once in Sorrento you can keep the car or return it there since Sorrento has offices for several rental companies.

I have no advice on construction. That can happen any time and anywhere. However most of your driving will be on freeways, therefore the worst that can happen is the closure of some lanes along the way somewhere. I'm sure there will be some places where you will see that. Construction happens on American freeways too, that doesn't stop people from traveling. Just keep an eye on the orange cones.

On the GPS.
Regardless of availability or not, renting a GPS from the rental company will cost you 15 euro a day or more. If you have the car for 10 days, that amount will pay for a brand new GPS. In your shoes I would buy one in the States, if you don't have one already, load European Maps on it, and take it with you. You can also use the Maps feature on your Iphone5. However consider that using that will drain your Iphone batteries, therefore have an Iphone car charger with you.
Also I don't know about data usage for the Maps overseas (I usually use my Italian Iphone)

Using your cell phone as a GPS could leave you with a gigantic bill when you return due to the expensive data rates. We bring our Garmin Nuvi 2595 LTM with a current European maps SD card. It is very useful, but...it can take you on some interesting routes taking you to your destination. A good map is also very useful when your GPS takes you on unplanned field trips.

We use a combination of cars and trains to meet our transportation needs. A car in a large city is not a benefit because of expensive parking rates, difficulty driving in cities, and the ZTL's. The previous poster gave you cogent suggestions to follow.

Once you are in Sorrento you may be tempted to drive the Amalfi Coast, which is spectacular. We have driven it three times and only my wife was really able to enjoy the scenery. The first time driving I maintained a white-knuckle-death-grip on the steering wheel. The road is NARROW, windy, and intense. it would be easier to take the bus and get off and on where you want.

Thank you. This is great information.

There are many "offline GPS navigation apps" for cellphones (Android and iOS aka iPhone). They are heavy (up to 2GB), but they load all maps into your device and don't require any cellular data traffic to work. They will work even if your smartphone doesn't have a SIM Card or has all cellular traffic disabled.

So they are like a GPS unit, only that you don't need to carry an extra gadget. They have the same functions a regular GPS navigation unit would have.

Garmin and TomTom have these apps for smartphones. They cost something, but are good value in my opinion.

I used my Garmin GPS with a Garmin Italy map SD card on my last trip to Italy, driving from Venice to Cinque Terre, then to Orvieto, then to the Amalfi coast, then Sorrento and finally to Naples airport. I pre-programmed some destinations (e.g., parking garages) before I headed off on the trip. It was great, and I never used a map for the entire trip. I've used this strategy in Germany/Austria, France/Belgium, and Spain as well, and it helps a lot. Except once, when the GPS guided me up through a very narrow castle road as a short cut to get to my hotel in Salzburg. It's a good idea to explore areas and routes using Google maps before you head off on the trip so you can avoid letting the GPS lead you along blindly.
But, on a more recent trip to France/Belgium/Netherlands, I also used my iPhone with two apps that used pre-loaded maps, and I highly recommend them: MapsWithMe and CityMaps2Go. You can download the apps and some maps for free, or pay a few dollars if you want more maps or the enhanced apps. As mentioned by Andre (above), the GPS on your iPhone will work for free and you don't need any cell data service to use these apps. In fact, make sure that you turn off cellular data before you travel! (Unless you have an Italian cell phone.) These apps were particularly useful for guiding us when walking around Paris and Amsterdam -- it's better than using a paper city map when you can also see where you are on the (iPhone) map!

Thank you. This is good information.
Robert

Regarding train vs car rental, I've compared these options on multiple trips, and for us (two couples traveling together) the car rental has always proven to be less expensive (per person), quicker, and less complicated than traveling by train. It's been cheaper than train travel even when we've had substantial one-way car rental drop-off fees. And don't forget about the time 'wasted' going to the train station, waiting for the train, and waiting for connections. But if you are just one or two people traveling, the car rental could certainly be more expensive overall, and maybe for some routes the train is a less expensive option even for two couples. You need to check the car rental rates and compare with the train fares for your routes. Also, admittedly, some people don't want to drive in Europe, but for me it's a great part of the adventure and it's great to have the flexibility. And some people simply prefer to travel by train.

If you have an Iphone5 to disable cellular data do the following:

Go to settings.
Click on 'Cellular'
Ensure "Data Roaming" is off (gray). It should be off by default. If it's green, tap it to switch it to gray (off).
Tap "Cellular Data" off (switch from green to gray).

Now your smart phone has become a dumb phone. You won't have internet access, but you can still make and receive phone calls and also send text messages.

The Iphone Maps feature will not work, but, from what I read above, there are some GPS Apps that you can load that work also with the cellular data off. I've never tried those, but I'll look into it, so that I don't have to carry my Garmin next trip to Europe.