Sorry to disappoint you, but "sights" will be tourism. There's nothing wrong with that! One of my most successful lines in Italy is "Essere turista è un duro mestiere" -- Being a Tourist is Hard Work.
That said, if I were in your shoes and truly wanted to get a little feel for non-tourist Italy in 9 days, I would rent an apartment in a town or village, of not more than say 6,000 inhabitants, somewhere near Florence but not too near, not one of the more popular tourist places (i.e.,
not Montepulciano, Montalcino, Pienza, Fiesole, Castellina, etc.), and not a villa in the country, but a small apartment right smack in town, preferably on the ground floor. (Florence is quite immaterial, actually, and any small town anywhere in Italy would do nicely; but I'm allowing you a day off

to go snap pictures of Florence.)
I'd also prepare for my trip by getting the essentials of the language down; you're running very late for that, mind you: in a month, it can be done, but it's unlikely.
I'd spend a lot of my time shopping. I would shop every morning, not just once at the beginning of my stay; and if in a shop I saw the old ladies all ganging up on some particular thing, I'd buy it too, asking them what they do with it. You'll often start a lively exchange right in the store, how Anna Maria makes it, but Madonna, the best was when Celestina, Sara's grandmother who died a few years ago, used to prepare it for the parish festa. You may wind up cooking it, you may wind up being invited by someone to try it the way
it should be prepared -- people are often very proud of our own cooking!
The rest would be mostly window-shopping of course, at grocery stores, clothing stores, hardware stores, stationery shops, etc. I'd watch television. I'd make note of the weekly market days in my town and others near it, coöpting your landlady or the postman's wife or another to let you accompany her on her rounds.
I'd attend Mass on Sundays (many places also have Rosary recitations and devotional services, often around 6pm; if you're a devout Catholic, go). Even if you're not Catholic, though, if you sing, you could prepare a simple Negro spiritual or other piece of religious music, theologicaly neutral of course, showcasing more your country than you (and easier to pull off!!), and on arriving I'd go see the parish priest, and tell him that you're in town not to see sights, but share just a tiny bit in the life of the community, and this is your little gift: I bet that would be a door-opener, churches are always scrambling for talent.
I would ask to visit a grade school; the odds are the school would be delighted, you'd learn tons of stuff, and your visit might be truly memorable.
If you have even a perfunctory interest in history, I'd ask, on arriving, who the local "studioso del paese" is, the guy -- usually a man -- who knows where the tombs and ruins are, and often keeps a scrapbook of photos, plus will be bursting at the seams to show some new person around what he knows and loves. (His fellow villagers you see have heard it all; they're grateful to him for keeping history, but may think he's a bit nuts. You, on the other hand, are a fresh audience.)
You could also, on arriving, or in advance, get in touch (via the Tourist Office, the Italian counterpart of your professional association, or the Rotary Club -- which is profession-based) with someone in town who is in the same line of work as you, and take it from there.
In a small town, a courtesy call on the mayor is not out of line. Firmly resist the temptation, however, to ask the mayor for the slightest favor -- no opening doors for you -- unless just maybe you had a very specific request in some field you're completely informed about, as for example genealogical records of your family. If she volunteers something, take her up on it, gratefully, whether you think you're interested or not: you probably will be (or should be).
At the end of my stay, I'd give any helpful people a farewell dinner in their favorite place -- not yours. I'd also take with me a few hostess gifts from my part of my country, something typical and likely to appeal; for the people in whose homes you wind up having dinner, or who may have squired you around.
Footnote on cameras. The purist would advise no camera at all; you're building memories of a cultural experience, not a record of a tourist visit. More reasonably, I'd take a camera, but not wear it all day (not in church or while shopping, for example, nor in people's houses -- although you might have it in your handbag in the latter case); and absolutely, absolutely, do not point it at people without their explicit permission, or, best, encouragement. A good telephoto lens or one of those 90-degree thingies will let you get away with a distant candid from time to time, but all in all, maybe even better not that.