12 days is not a lot for those three locations, it is 'doable' but you'll have to be selective as to what to see. The problem with looking at itineraries that organized tours do is that things take longer 'on your own'. I think they are better 'on your own' cause you get to decide what to see and how long to stay, but you'll probably find there is so much to do in each of those locations you'll wish you had more time. Given that, I would limit yourself to three 'bases' - Florence, Sorrento, and Rome with possible day trips from them. I would probably split the time equally with 4 days in each (this assumes you have a 'real' 12 days, that the 12 doesn't include flying days, if it does include those then you really have only about 10 days to actually see anything). Rome deserves at least 4 days with no day trips. Florence you could easily do one or two (if you have 4 full days) for day trips. The most popular ones are Pisa/Lucca (one day) and Siena. From a base in Sorrento you could do one day to Amalfi/Positano/Ravello (try to take the boat one way and the bus the other for best scenery options) and another day either Capri or Pompeii.
In Italy there are tons of small, family run hotels, I wouldn't limit yourself to Air B&Bs. The small hotels are just as interesting, in many cases nicer, probably about the same price. It's not like in the US where hotels seem to be either chain hotels or expensive high end places.
Depending on where you arrive/depart from you should put the departure city at the end to avoid having to change hotels one extra time. So if you arrive in Florence and depart Rome you would do: Florence, then Sorrento, then Rome.
Here's my photos of Italy - https://andiamo.zenfolio.com/f739967755