Where to stay in Florence

I'm familiar with all 3 areas. Campo di Marte and Oberdan will have the most local feel. I'm partial to Campo di Marte (and adjacent Le Cure), with the only downside being that it's a 30 minute walk into the historic center (longer if you're behind the stadium area) and so you'll probably tend to take one of the frequent buses when you go. Oberdan is closer to the center but still has a local feel. I'd avoid anything very close to the ring road, which tends to be noisy due to heavy traffic, or that faces the railroad tracks. Sant'Ambrogio is very lively with a great market and lots of shops, but lively comes with noise and crowds. I stayed in an apartment on Via dei Macci last year and wouldn't stay there again, but something closer to the ring road might be more tranquil.

Which/Where is your school exactly?
Sant'Ambrogio is part of the historical center in the district of Santa Croce. In terms of cafes and market (Sant'Ambrogio market is here) this area probably offers more.
The area of Piazza Oberdan is very residential, upscale too, but not too much in terms of restaurants and cafes or markets. Although you are not too far anyhow and there are many commercial establishments (supermarkets, restaurants, etc.) on nearby via Gioberti.
Campo di Marte is also primarily residential area (also more upscale than others), with lots of sports facilities (stadium, Mandela Forum, Costoli Swimming Complex, etc.). There are a few restaurants here and there, and markets are available once a week on Piazza delle Cure and also closer to the stadium.
Both the area near Piazza Oberdan and Campo di Marte, being residential, are quieter in the evening than the historical center (Sant'Ambrogio). The advantage of Sant'Ambrogio is that you can easily walk to the center, from Piazza Oberdan is a longer walk, and from Campo di Marte, it's a very long walk, so you'll need a bus (or a bike).

Thank you both for your reply. I hope to go to the Angel Academy at via Nardo Cione 10. It is about a 9 minute walk from D’Ambrogia. I may be in class in the evenings and don’t want a long walk home. However, I don’t want to stay in a noisy or touristy place. My budget is between €1,000 to €1,400 monthly. Is this a reasonable budget for a three room apartment with a kitchen, WiFi, air conditioning, and washing machine? Thanks again.

If you can find an apartment in the area of via Gioberti/Piazza Oberdan, it would be better. The walking distance would be short and it's a nice area. There are plenty of places to eat around via Gioberti and there is a Coop Supermarket.
Properties in Sant'Ambrogio are older and smaller, yet more expensive because in the historical center. The area of Piazza Oberdan developed in more recent times (mostly in the late 1800s, which is basically yesterday in Florence).
3 bdrm apartments would be considered a rather large dwelling, so the choices will be fewer. If you can do with 2 bedrooms, you'll have more choices, especially if looking for a furnished apartment. Prices in that area of Florence, for a furnished apartment, should be around 1000 euro a month for a 2 bedroom, 90 sq mt apartment.
If you absolutely need a 3 bedroom apt, you are looking at about 110/120 sq.mt. and you'll be approaching the upper limit of your budget of 1400 euro/month. However if you go through a vacation rental website (catering to foreign tourists), they'll charge easily 800-1000 euro a week for a 2 bdrm apt., so you need to go through a website for long term rentals.
Obviously in the 1800s they didn't have air conditioning, but most people have installed compact units like the one pictured here above the bed. They are often installed above the windows in every room. The number of bathrooms are also rarely as many as in an American house. In a 2 bdrm apt there is usually one bathroom only, while in a 3 bdrm apt there are no more than 2. Washing machines are usually placed either inside the bathroom or in a small laundry closet. Wifi will depend on the landlord.

The websites for rentals used by Italians are:
www.mioaffitto.it
www.affitto.it
www.casaclick.it
www.bakeca.it
www.immobiliare.it
and others.

Not sure if any of the above has an English version. The italian word for furnished is AMMOBILIATO
Italians do not count bedroom but VANI or LOCALI (i.e. rooms) which thefore include more than just the bedrooms, but also living room and dining room. So for a 3 bdrm apt you need to search for a 5 vani or locali, while a 4 vani (or locali) will be a two bedroom place.

Vacation rentals can be found on www.homeaway.com or www.homelidays.com and others. They list prices per day or week, and since they are very short term rentals for tourists, they are really expensive (a 3 bdrm is probably at least 150 euro a night)

Roberto, Mille grazie! If I go for two months I will rent a 3 room apartment. However, if I go for a year to three years I will rent a four room apartment. Hopefully, my daughter will come for a visit. The Italian sites are less costly than the Hoilday rental. I wonder if the Italian agencies will rent to me for only two months? I will definitely call them for a long term rental.

Many properties in the websites I listed are unfurnished (vacation rentals are all furnished of course).
If you rent an unfurnished place for a long term rental with the intention of furnishing it yourself, you must know that unlike in America, an unfurnished apartment in Italy means "really unfurnished". That means there are only bare walls (and sometime not even the wires). In the kitchen there is absolutely nothing, no sink, no stove, no fridge, no cupboard, no dishwasher. Just a hole in the wall to connect the sink disposal to the sewage system. Same with the light fixtures, there are only live wires hanging from the ceiling, they don't even leave you a bulb. In the bathroom they generally leave the toilet and the sink, but that's only because they are cemented to the floor and too hard to pry off, otherwise they'd take those away too, but you'll need to buy the toilet seat yourself, because if it can be unscrewed, it won't be there.