Rent a car through a consolidator, such as:
www.kemwel.com
www.autoeurope.com
Both are part of the same company based in Maine, but price vary so shop both. They work with the largest international companies operating in Italy (e.g. Europcar, Hertz, Avis-Budget). Generally Europcar has the cheapest prices, once you consider the price of no deductible insurance. Avis is the most expensive.
Make sure you select the No Insurance Excess (no deductible option), by clicking the box in the above sites. I've had issues (i.e. unwarranted after the fact charges which I had to dispute with the credit card) with all of the above, including Avis/Budget which charged me for fuel charges in spite of having returned it full. So don't assume anything. Just take a picture of the fuel gauge and keep receipts. Make sure the attendant marks the receipt "full tank" when you return it. The bill always comes a few days later via email, so expect surprises like that. Check what you have been charged.
As mentioned above, speed is checked by Autovelox machines with cameras, therefore don't assume you can speed just because there are no cops around. Of course locals speed because they know where the cameras are, but you don't.
Historical centers are closed to traffic, except for residents and transit vehicles. Google: Zona Traffico Limitato to see what they are and what the signs look like. Entry in a ZTL is prohibited to those without permit, and it's enforced by cameras. They are well marked in spite of what people say in this site.
Also review the Wikipedia article on "Road Signs in Italy". Italy uses the International Signs used by the rest of Europe.
My father owned a Punto most of his life and I'm familiar with it. It's nowhere close to a Ferrari, but it's better than a Panda, although for two people and a half like you, a Panda would do also.
In preparation for your trip practice parallel parking in very tight spaces. Also find a roundabout near you and practice those too. Nearly all intersections in Italy (in Europe actually) have been converted from street lights (or stop signs) to roundabouts.