Ciao Margaret,
Being a EU citizen will make everything much easier, as you can settle without having a need for a visa beforehand but you will still need to go through the process of getting a permit to stay "permesso di soggiorno" as an EU citizen so prepare for the bureaucracy by reading ahead what steps you'll need to take.
As far as where to base yourself: are you staying in central Florence this July? I would suggest you take some of those days and take day trips to check out Pisa, Lucca and Siena while you're here. They are major towns but still smaller towns compared to Florence which would make good bases. They tend to less costly than Florence (although staying in any historical center will not be cheap) but they will give you easy access to trains to move around while you live here.
There are definitely smaller towns along the train lines you can consider, it depends how much of a small town experience you are looking forward to living. Whereabouts do you live in California and how much of a change are you looking forward to? It might make sense to not go from city to small village at first, until you get a chance to be here and move around and see and experience how locals live. Then if you decide to stick around beyond the year, you will know where you want to go.
As for a private room with all expenses and services included, the bigger cities will offer more choices as they are used to tourism and providing those types of lodgings. But what you want is a studio apartment since you're interested in a kitchen, and most B&Bs that offer rooms will not offer kitchen access.
So I suggest you start looking at small apartments found on AirBnB, for example, to see what's available. Consider that in larger towns where people rent apartments by the week, they will be used to getting that weekly income that by the month might not be convenient so in that case the smaller towns along the train line might work better - for example, Empoli, Lastra a Signa, San Miniato are some towns between Florence and Pisa, or Montecatini or Pistoia on the line toward Lucca.
Hope I've given you some things to consider that will help as you continue your planning! I look forward to hearing back from you
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