I am currently living near Pisa, and have spent time in Rome and other cities. The free Wi-Fi situation is... variable. There can be issues and some setup required, but you shouldn't have too much trouble.
Many cities and large shopping centers do offer free Wi-Fi services (There is something called "Wi-Pi" for most of Pisa, and the project you linked to is another example). Most do require some form of verification. This is most often a text message or credit card number.
The text message option, as far as I can tell, only works for Italian phone numbers. I tried to use this option using my US cell number and my Google Voice number, and neither was successful. You could of course get an Italian SIM, but that may defeat the purpose of the whole "free" aspect.
As for the credit card option, it is usually just for identity verification, and will not actually charge you. Be alert, however. Some of them will charge for usage over a certain time or data amount per day, though the amount you would need to use is usually quite large and I haven't seen this too often. Just make sure you know what you are signing up for.
If you successfully create an account on one of these services, the cool part is that you can use that account at any of the hotspots that are on that service - One that I have an account for works at my local supermarket, shopping center, the airport, and a couple other places around the city.
If one of these larger networks is not in the area, or you can't do the verification, many coffee shops ("bars" in Italy) will have free Wi-Fi available for customers. So you can just stop in for a coffee and ask for the Wi-Fi network.
(Un)fortunately, Starbucks is not allowed and therefore does not exist in Italy. McDonald's supposedly has Wi-Fi but I couldn't get it to work - and nobody there was too keen on solving my problems. Fast food places are also somewhat scarce but in the larger cities you can find them.
Lastly, keep in mind that any of these networks, especially the smaller ones in individual shops and such, may not be functioning that day or may have issues etc. (Italy is notorious for slow bureaucracy and poor infrastructure maintenance. I will even lose cell service inexplicably for hours at a time about once a week).
So if you can't connect or start having problems, you might need to either wait it out or look for a new network. I think this will be somewhat less of a problem in the big tourist cities though.
Speed overall is quite good actually, when the network is functioning, but the larger networks depend on how many people are using them, so I have actually had better luck speed-wise with smaller networks in coffee shops and the like. These will also have less security measures in place, which can be good if you want to download stuff or view content through protocols which might be blocked on the larger networks.
Tl;dr: There is Wi-Fi all over the place in big cities but it often requires validation, payment, or can be a pain to connect to. I would try coffee bars first, especially those that advertise Wi-Fi.