If you want ideas on locales to base in, it would be helpful to have more information - for example, what countries potentially interest you? Your list of interests is general enough that it could be met in dozens of places. Can you be more specific?
Florence is a good start - lots of great places that are reachable from there for day-trips. If you're going to be there a month, I assume you'll want to rent an apartment. Even with an apartment rented for a month, you might want to spring for an overnight or two in Rome and an overnight in Venice - it's possible to day-trip there, but there is so much in Rome that you'll likely find a day-trip too limiting, and Venice is at its best in the early morning and in the evening when the day-trippers and cruise passengers are gone. You might also want to look at Bologna, which has great train connections to a myriad of places, including Florence, but is less dominated by tourism.
You say you don't want to rent a car, and that makes sense if you'll be spending your time in places well served by trains and buses. However, there are fabulous countryside areas that can really only be explored by car, for example the Val d'Orcia region of Tuscany (Montepulciano, Montalcino, Pienza). You might consider renting a car for just two or three days to visit areas like that.
As to when, generally you will find shoulder season a good mix of not-too-crowded and pretty good weather. My personal pick is spring because I like the longer days, although I think you may get more rain in spring than in the fall.
Find a debit card that doesn't charge a percentage fee for converting and doesn't have charges for using ATMs other than your financial institution's. I have Schwab, which charges nothing for withdrawals anywhere in the world and rerfunds any fees the other machine charges, plus they have awesome customer service. You also want a credit card that doesn't charge beyond the standard 1% - Capital One is one possibility, but there are others. Call and ask. For sure, before you go call your banks (ATM and credit card) and let them know when and where you'll be traveling so they don't deny your charges as possible fraud.
My T-Mobile plan has unlimited data and text from Europe, and free wifi calling, so it costs me nothing to call from the hotel or apartment where I have wifi. The per-minute charge for calling not on wifi is 10 or 20 cents per minute, can't recall, but nothing that would stop you from making a short call home while you're out and about. It is possible to get a cheap phone and phone plan in Europe, which others can tell you about, but it would depend on where you expect to be calling. If you think you'll be calling mostly other places in Europe, that might make sense, but most of my calls are home, and I don't really feel constricted in sticking mostly to wifi calling.