mk,
The first thought that occurred to me when reading your post was that it would have been prudent to have worked out the details before booking flights and hotels. I'm also a bit confused about your Itinerary. Are you planning to stay in Montecatini and then take day trips to each of the locations you mentioned?
Given the very short time frame and the fact that flights and hotels are already booked, that tends to limit the possibilities somewhat. It would help to know....
- when is this trip taking place?
- where are you flying from?
As you are not "experienced overseas travellers", you may find that you'll be affected by jet lag, so won't be up to full touring speed for a day or so.
Travel by train will likely be the best option as it will be much faster and more efficient, especially on the high speed routes where the trains travel at up to 300 km/h. However there are some potentially expensive "caveats" to be aware of when using public transit in Italy. If you need more information on that, post another note here.
A car rental may not be the best idea. It's important to note that for driving in Italy, each driver listed on the rental form must have the compulsory International Driver's Permit, which is used in conjunction with your home D.L. (so you must carry both). These are valid for one year, and easily obtained at any CAA/AAA office (two Passport-sized photos required, which may be provided by the issuing office).
You may also want to have a look at some of the posts on the forum concerning the dreaded Zona Traffico Limitato (limited traffic) areas that are becoming increasingly prevalent in many Italian towns & cities (especially Florence!). Some of these are enforced by automated cameras and some by local police. Each pass through one of the automated Cameras will result in a €100+ ticket and visitors often don’t know of these violations until several months after they’ve returned home. In addition to the actual fines, renters will also be charged by the rental agencies for providing information to the authorities. You may find this website helpful - http://www.slowtrav.com/italy/driving/traffic_cameras_speeding.htm
There’s also the possibility of fines for driving in bus lanes, parking tickets, tolls and speed cameras including the devious Traffic Tutor system which not only monitors instantaneous speeds but also average between two points. Violate either or both parameter and expensive tickets will follow! Be sure to budget for the high fuel costs and note that some automated fuel pumps may only accept Chip & PIN credit cards.
A GPS along with a good Map is also prudent. Be sure to give some thought to the question of CDW and theft insurance. In Italy, buying CDW from the rental firm is usually compulsory, so coverage provided by credit cards will not apply. Check the Car Rental guide at http://www.gemut.com/ for more information. Their free "Car Rental" PDF Guide can be downloaded. You may also want to check the “Travel & Transportation” pages at https://travel.state.gov/content/passports/en/country/italy.html for other tips. There’s also this website - https://it.usembassy.gov/u-s-citizen-services/local-resources-of-u-s-citizens/transportation-driving/?_ga=1.21563186.86862954.1474991758
I would highly recommend packing along a copy of the RS Italy guidebook as that has an enormous amount of information that will help you to plan efficient sightseeing.
If you could clarify the few questions above, I'll try to offer some Itinerary suggestions.