YES!! You can walk into any electronics store and buy phones even without a carrier, completely unlocked, to then be used with any prepaid SIM card you want to buy. You don't need a "codice fiscale" (CF) or Italian tax ID number to make purchases in Italy!!
Many of the electronics stores will be affiliated with a carrier so will also sell prepaid SIM cards as well but for the carrier they are in partnership with. For the SIM card you do need to show ID, a passport is generally sufficient, I don't recall a CF being required although it is likely requested automatically for registration.
There are many phone shops with the carrier brand outside downtown, almost one on every block - the main ones are TIM, Vodafone, 3 (yes, the brand is simply a number), and Wind. Those smaller stores also carry phones, but usually the latest models and not the cheapest. If you want more choice, I would recommend the electronic stores such as Expert, Euronics or MediaWorld. You say you are in Italy 5 weeks but are you in Florence the whole time? If you are, I can give you some more precise store recommendations, just let me know!