Florence Neighborhoods

The Oltrarno neighborhood is one suggestion - close enough to walk into the bustle of the city center but far enough away that you can leave it behind you in the evenings.
Another thought would be Fiesole - perched on a hill above the city but, again, far away from the crowds in the city center ... about 20 minutes from the city by bus.

All neighborhoods in the historical center could work.
Quietness depends on the specific street, not on the neighborhood. There are quiet streets in busy neighborhoods, and also noisy streets in quiet neighborhoods.
Outside of the city center, you will have to deal with traffic and lack of English.

We loved our 5 weeks in an apartment on Borgo la Croce near Piazza Beccaria. Pedestrian only street lined with coffee bars, restaurants, and regular stores like hardware stores. Just inside the old city walls. Quiet.

A couple minutes walk west to the Sant' Ambrogio market and a couple minutes walk east to a pretty large and good supermarket. One minute walk to the bus stop for the little electric buses that can take you almost everywhere in the historic district. One and a half minute walk to what, after much testing, we thought was the best gelato in Florence, Gelateria De' Medici. About 15 minutes walk to the Duomo.

The helpful and English-speaking airbnb superhost has two apartments in the same building, one on the top floor, Le Rose Siberine (ours) and one on the floor above street-level, Le Rose di Leonardo. She lives in the building, too. We would stay in either very happily.

We have also stayed in the Oltrarno for a week and enjoyed that, too, but it was considerably busier and noisier. Of course, it depends on the exact apartment you choose. If we stayed across the river again, we'd probably stay in the San Frediano neighborhood because several of our favorite places to eat were there and it would be near the British Institute where I like to take history of art classes.

Thanks all for such great information. We have done long term stays in Spain which were super. The street we are considering is Via Della Colonna.

Although nearly the entire historical center is closed to non resident traffic, including Via della Colonna, however that street is a major bus route, therefore if the apartment is facing the street, you may be subject to busses’ noise and fumes.

Via della Colonna is fairly commercial (museums, schools, businesses) until you get out towards the park and the Viale (ring road). There's a Carrefour grocery store nearby but it's pretty small, and the much nicer Coop and Conad City stores are a somewhat long walk with groceries. I've stayed in 2 apartments near that street and both had some recurring loud noise.