Rent a car in Florence. Getting out of Florence is pretty straightforward. There are rental locations in town near the train station & at the airport.
Getting TO FLR airport is easier now with the new tram line, but the rental car center is offsite. You could take a taxi directly to the rental car at the airport. From there it is very easy to get onto the road to the Chianti region - did it 3 times in the last month.

Chianti is a pretty big region, and not just one town. Options include staying in small to med sized walled towns in or near the Chianti area, staying in a bigger city such as Siena, or staying in the countryside at an agritourismo or podere (farm house), or some resort property. Depends on your style, budget, etc.

When are you going? If it is in summer then you'll want someplace with a pool which means outside of a town/city - it'll be 100F there this next week and the heat in the afternoon can be oppressive.

Here's info i gathered about renting car in Florence/Italy.
Italy car rental options - what would you do?

Driving from the airport's offsite location is the best option. personally, I'd head to 222 and enjoy the drive.Driven and stayed in Chianti at least 7/8 times. Easy driving and pretty country. We'll return this Sept. enjoy your planning.

Reverse the stay---get your at the airport and do Chianti first, then finish in Florence.
I would split my time in Tuscany and spend time further south for variety.

Intenational Drivers License mandatory in Italy - get at AAA for about $20.

Whether you're asked to show it or not, unless you hold a European Union license, Italian law requires drivers to carry an International Driving Permit in addition to your state-issued license. Not all countries have this law, but Italy, Austria, Greece, Poland, Croatia and a few others do.

https://www.autoeurope.com/internati...-permit-italy/

Some rental agencies don't ask to see an IDP, even though you're obliged by law to carry one if you don't have an EU conformant license. More and more agencies are checking that you have one, and I read one post on TripAdvisor by a person whose vacation was rather riuned because the agency refused to rent him a car without an IDP.

I've been asked for my IDP when picking up rentals in Tuscany. I agree with Bob about splitting your time in Tuscany between Chianti and somewhere further south in Tuscany (Val d'Orcia area).

There are very few places in Chianti with train service, but it depends on where you are staying. You might find bus service, but even that is limited. I would expect any rental car companies to be very limited too.