those are perfect months for visiting Tuscany
Ciao bronwyn,
Before being able to offer specific suggestions, do you already have a preference whether you would prefer to be in a town such as Florence or Siena or out in the countryside?
Are you planning to rent a car at any time or are you thinking of depending solely on public transportation?
The train system works very well but for it to be easy for you to manage 1-2 months here you'd definitely have to consider staying within a town - whether it is large or small - and not in the countryside a few km or miles away from the train station. You could consider one of the smaller towns such as Castelfiorentino or Certaldo to the southwest of Florence or Pontassieve or Rignano sull'Arno to the east of Florence - these are all along train lines. Pisa and Lucca would also make excellent small towns to stay in, both have small city centers and would make for a relaxed stay with the convenience of having the rail system at your disposal.
To get your search started, this agency has
apartments in Florence available for longer stays and not just for a few days or a week. You definitely want to contact them directly to ask for a price estimate of an apartment for 1-2 months for that time of the year, keeping in mind that what you see on the site is aimed to short stays.
Staying in
Florence would be very convenient - you could stay out of the historical center and settle in among residents. You can easily get downtown and to the train station to get everywhere, including down to Rome, Naples and the Amalfi coast.
September generally is a great month, with sunny days and mild temps. October is that month where temps start changing - you'll get some nice sunny days mixed in with rainy days and cooler evenings. The period would be excellent as the high summer season winds down.