Budget Car Rental in Florence

We always book our European car rentals with AutoEurope.com- a US based consolidator with excellent customer service and usually the best rates
Have used Hertz and Europcar thru them, those 2 seem to be the most prominent agencies in Italy

Before you book with Budget you might check with AutoEurope.com
Their website is straightforward or you can call their toll free number, they are extremely helpful and knowledgeable

What exactly are your concerns?
It appears that Budget has 3 locations in Florence, including one at the airport. A quick look online is showing average to poor reviews for their customer service.
You might consider taking the train or bus from Florence to the general area you're trying to get to and then renting there - will be a much more mellow experience than trying to negotiate Florence city traffic getting out of the city.
Concur with ChristineH regarding AutoEurope for all of the reasons she states, adding the fact that they provide a simple and straightforward explanation of the various insurance options, which can be a stumbling block when dealing with most rental car companies directly.

We used Hertz's office on SanSovino. It's about a 2 mile taxi ride from the train station.

They have maps giving you directions out of town without getting close to the center city.

Tuscany is best seen from behind the wheel of a car.

We rented from Avis at the Florence Airport location 3 weeks ago through Autoeurope. Avis/Budget are the same counter and staff. They don't receive great ratings but we had no issues at pick up or return.

I also use AutoEurope.com.
Budget is part of Avis, therefore the Budget rental will be in the same shop as the Avis rental.
Below is the Google Map Streetview photo of the Avis/Budget office downtown Florence on Borgo Ognissanti, the street where all rentals are. The cars are stored in the Parking Garage to the left of the office you see in the photo.
https://goo.gl/maps/NRnAJCPjpvxGpqTm8
I've used it once only in Florence. No complaints.

Avis/Budget has another location at the airport, in case you need to rent upon landing.

Just returned from 2 weeks in Italia. Rented from AutoEurope in Florence. Took pictures of 2 damaged areas of the car in the lot prior to leaving. Good thing I did. When I dropped off the car at a different location, they inspected the damage...but the pictures from the rental lot pretty much ended the conversation. I also paid for total coverage, also a good move.