Better to do the weekend in Florence or Chianti?


We will be training from Rome into Florence, then drive to Chianti, then drive/bus/train to CT. This is in mid May.

We have 5 nights for Florence & Chianti region. I was thinking of doing 3 nights in Florence and then 2 nights in Chianti. I just realized however the 2 nights in Chianti will be a Friday & Saturday night.

My question is, is it better to do the weekend in Florence or Chianti - is there a difference? What is your recommendation?

In Florence, we will be mostly sightseeing, eating, shopping and just chilling in the city. In Chianti, the plan is to visit Siena, San Gimi, and wineries.

If I could sneak another question in, would you also recommend a visit to the San Gimi thermal pools?

Please advise, thank you!