That isn't a lot of time to fit in all that you want to see, the first one I'd skip would be Cinque Terre to have more time for Tuscany.
First, are you willing to rent a car to see Chianti? There isn't much public transport to make it quick to visit, so if you have a car, I'd suggest you arrive, quickly see Pisa and then move on to the Chianti area. From there, you can drive around and go to Siena.
If you're not planning on driving, then I would suggest you use Florence as your base for all of your nights. It will make it easier to take a bus from there and go to San Gimignano, and then on to Siena. You will see Chianti as the bus drives through it, but not really stop in it UNLESS you join a tour - take a look at the many ones offered through Viator:
https://www.partner.viator.com/en/8357/
Even if you've been to Florence before, it is the best base when you have no car.
The second alternative would be to head on to Siena directly and stay there. There are buses as well as tours that depart from there.
You'd need to take a bus from Pisa, passing through Florence before getting to Siena - more details here:
http://www.discovertuscany.com/siena/tourist-info/getting-to-siena-without-a-car.html
Now if you really really want to fit in CT in, I won't say you can't do it and offer the way to do it:
fit it in the first day when you are in Pisa (since you're closer). Take the train up to Riomaggiore, take the ferry from there to Monterosso to see all the towns from the sea and then use the train to move in between them. The trails along the sea are still mostly all closed, so the easiest way to visit is by train.
In this case, I'd sleep there overnight the first night before heading back into Tuscany - and at that point, I'd suggest you stay in Siena, seeing as that is really more what you want to see than Florence!
If you like any of these plans, start looking at hotels and accommodation so that you can pick something nice rather than be left with few choices. It is already a busy time in CT and Tuscany so you can't wait until the last day to decide where to stay, otherwise you will be worrying about that rather than enjoying your travels!