Florence/Siena

My suggestion is to spend your time in Florence - it's beautiful and full of great history, art, culture, and good food. I wouldn't rush away, particularly if you don't even know what you're rushing to.

A one-day car rental is just not efficient, given the time involved in picking one up and returning it.

I agree - stay in Florence. It will be very crowded in June; but so is Disneyland. As is Disneyland, Florence is a famous destination for very good reasons. Get advance tickets for your choice of attractions. If you must see Chianti, take the tour. Avoid spending your time moving from place to place. Two nights in Siena is only one full day plus 2 part days (at most) and the town deserves all of that time and more.

You might look into Tours by Roberto, which generally included wineries, countryside, luncheon, etc. Roberto is a favorite of Rick Steves and appears in his videos of Tuscany.

Last May/June we spent a month in Tuscany on an anniversary trip, splitting time evenly between the countryside and Florence. In retrospect we now wish we'd spent more time exploring the countryside and less in the city. As much as we love Florence, which has long been one of our favorite cities on earth, the heat and the crowds got to be unbearable for us.

My advice therefore is contrary to the others. Suggest renting a car as you depart Florence and driving the very pretty Chiantigiana Road down to Siena. Allow most of a day for the drive which, while scenic, is slow moving and meandering. Not that you'll particulary care - it's worth the time.
Personally, I'd stay in Siena an extra day even if you have to carve an extra day from Florence to do it - one day devoted to the Chianti drive and to enjoy the town and the other to take advantage of the freedom and mobility that a car provides to explore the Val D'Orcia and the Tuscan hill towns to the south. Staying over in Siena will allow you to enjoy the city in the early morning and late afternoon without the day trippers that can clog the center of the old city. It's a completely different experience without the crowds.
During our own visit to Siena last year we had no problem finding parking in the underground lot at the train station, but do check that you won't be arriving during a major festival when parking anywhere can be frustratingly scarce. Steering clear of the weekend would be a good idea for that reason.
No matter how you organize this you'll have a great trip, but our personal bias is for Siena and the Tuscan countryside.

Thank you for your replies and suggestions. I like the idea of renting a car on our way out of Florence heading to Siena and driving thru Chianti... Any tips of convenient places to rent a car in Florence? Our Hotel in Siena has parking available.

Renting a vehicle on the way out of Florence is a frequent topic here. Suggest using the search feature at the top of the page to read some of the recent discussions. There are several very informative threads there that are worth reading.
Based on the advice we received here we opted to book thru AutoEurope for our own trip. The rental was thru Hertz, and there was a handy location on Via del Sansovino that we used without too much drama. It's well outside the area where ZTL's are a concern and getting on the road south from there is pretty straightforward. There are other locations however, including pickup points at the airport, so do study a map - this is one subject in particular that's covered very well in previous discussion threads as mentioned above.
One more thought: driving into Siena itself can be problemmatic due to the ring of ZTL zones that surround the old town. You may want to confirm with your accommodation that they're either outside the ZTL zone or that guests are allowed to enter without penalty. Can be an expensive mistake if you get it wrong so by all means confirm it in advance.