Ann,
In my experience, it's doubtful that you'll be learning to "speak Italian first-hand" in a one month trip, unless you're attending school. However, being in the country is certainly the best way to learn.
The most economical place to stay for a month would probably be a Hostel, but I haven't priced Agriturismo's. For example (since you mentioned Under The Tuscan Sun), the San Marco Hostel in Cortona has dorm rooms for 14€ per night, which includes breakfast.
If I read your post correctly, you're planning on staying in the same place for the entire month, perhaps just venturing out for day trips? Since Agriturismo's are usually a bit outside major cities, you'd have to consider transportation options as well (unless you were also going to hire a car for the full month).
Regarding the aspect of "an extended trip...alone?" on my last mega-trip in the fall of 2006, I backpacked through Germany, Italy, Switzerland and France for two months. That sort of travel kind of "forces" one to be a bit outgoing, and I usually seemed to have lots of people to talk to (I met some interesting people!). A couple of my stays were in Hostels, which of course inherently provides "company".
However, I discovered on that occasion that two months seems to be my "limit" - towards the end of the trip, I was starting to feel that it was time to go home and see my sons (and of course the dog!). Perhaps it was just the realization that the trip was indeed over and the travel funds were becoming depleted?
I'm planning another mega-trip hopefully for next year, and if it does happen it will be interesting to see whether my reaction is the same as last time.
Good luck and happy travels!