Florence

Personal preference, maybe, but I'd stay in the center as, aside from any day trips, that's really where all the attractions are. It's also nice to be able to walk around in the quieter early mornings and at night for some different perspectives? LOL, we cut a rug to a really good jazz ensemble on Piazza della Repubblica one night, Piazza della Signoria is at its Medieval best after dark, and the polychrome detail of the Duomo complex is even more impressive when it's all lit up. Florence from up on Piazzalle Michelangelo at night is pretty darn beautiful too.

Florence is not as sprawling as you might think; we were there for 5 nights and walked everywhere, even from train station to hotel, which was probably a little further away than many would have done it.

Have you considered renting an apartment? On our last trip to Florence we stayed here:
https://www.airbnb.com/rooms/1474636

Even though it was right down the street from the Duomo we felt far removed from the center once we entered our apartment. We walked everywhere and still talk about how cozy the apartment was. I developed a cold while there and the host had a nice supply of herbal teas that I drank most nights, which really helped. Love that we had a separate bedroom and living room, hubby cold go to sleep while I drank tea and planned the next day. We walked out our door and we were smack dab in the middle of it all with plenty of options for food and shopping.

Another vote for staying in the heart of the city.

I stayed at Il Salotto di Firenze, a B&B just steps from the Duomo. Even the largest rooms are well within your price range. And it was quiet and air conditioned.

One thing to consider is that if you're there in the late spring or summer, when temperatures can seriously climb, you will be grateful to have your lodging easily accessible so you can take a siesta in the hottest part of the afternoon, or even if it's not hot, just to go back and rest a bit between activities.

We are booking a place through airbnb. We have found that airbnb is an "affordable" way to stay in the heart of a city. Our apartment is on the 6th floor and has a view of the duomo!!!! We aren't staying until this summer, but we are excited about the adventure!

Our apartment is on the 6th floor...

Just a caution for anyone who might have issues hauling their luggage up that many flights of stairs? It's not unusual for older buildings in Italy not to have lifts. Be sure and check if that might be a challenge for you? Same goes for air conditioning if going in the heat of the summer.

Kathy is right!! When you rent an apartment read all the features. Our place had a lift, which was great (I think we were on the 4th floor?). Our apartment was so affordable for being in the center of the city. We are renting an apartment through Airbnb for our next stay but there will be four of us, so a different apartment this time.

Are you taking the train? We really liked the Santa Maria Novella neighborhood. Literally a three minute walk to the Duomo and many of the hotels have views of the Duomo and SMN from the rooms. It is also walking distance from the train station.

I would recommend Locanda Daniel, it's a lovely hostel, very close to the train station.