We started our official paperwork here in Italy by going to the "Tribunale Ordinario di Firenze" in the Piazza San Firenze. We were under the impression that this was where to start the process of getting our Permesso di Soggiorno, but it turns out we were wrong. There was a really helpful officer at the front door who spoke no English, but did his best to send us in the right direction. He told us that the process started by getting our "Certificato" in this building and then heading off to a Questura near Piazza San Marco for our Permesso di Soggiorno.
We followed his instructions and started with the "Certificato". This entailed going to the window at the end of the hall and filling out an application form - one for each adult, but the kids didn't need them. From there, we were sent out to a Tabacchi on the street to buy our tax stamps (called "bollo"). If my memory is correct, the certificate required about a 10-11 euro stamp, and we paid an additional 3 to 4 euros to have the certificate the same day. Without the extra "bollo" we would have had to wait 3 days for the certificate.
Finally we submitted the applications, the stamps and our passports at the same window and a few minutes later we received a stamped/signed official document of our "Certificato Generale del Casellario Giudiziale". I think this "Certificato" is actually some sort of record check - I'm still not sure if we were required by law to get it upon arrival, but it's nice to know that we've tried to do things legally.