Ciao!
To be honest, public transport - bus or trains - will work with your itinerary, but they are not great. I have looked for options but consider that if you don't want super long travel times and less stress about getting tickets and meeting a specific bus timetable/location, you might consider the following:
The easiest solution: car rental and drive yourself to these places.
Next best solution: car with private driver, but it won't be cheap.
Next best possible solution: bus where possible, train from Florence to Pisa is the easiest and then just try to run around and meet all the bus timetables because if you don't, there aren't many options after that.
Ok so let's look at each leg to try to help you figure out this last option, since it is the one you're asking about:
Pisa Airport to Volterra - doable by bus in less than 2 hours. Use the travel planner on
https://www.at-bus.it/it/
to get the details for all of the following, since that's where I go to figure out buses in Tuscany.
Here's a quick screenshot where you see only two options, you really want to take the
second one as it takes less bus changes and less time overall.
View attachment 47
Next leg -
Volterra to Florence: following day
View attachment 48
You could theoretically get the train from Poggibonsi to Florence but that leg of the bus ride is only 50 minutes which is actually better than taking a train that makes you change at Empoli... the first option above takes longer becuase there is about a 45 minute stop in San Gimignano while you wait for your next bus so it actually would give you a quick visit to the town.
I would say you need to catch that one since you arrive in Florence and that same evening need to get to get to Montecatini for the evening.
Florence-Montecatini
Now getting to Montecatini is "easier" because there are trains there BUT at the moment there is construction work on the line so there are no trains straight to Montecatini
. You need to take
a train to Pistoia and from there a
bus the rest of the way. Taking a look at the trip planner for the buses, to see if taking a bus the entire way makes sense, it actually doesn't - there are more bus changes and the trip takes longer.
So I assume the concert will be pretty central in Montecatini so searched for Montecatini Centrale and get these options. I don't know when you need to get there for the concert (9pm?) but I assume you'll want to catch some dinner first! So take some time to decompress and rest in Florence before catching one of these these two options:
View attachment 49
Arrival by 6.30pm could give you some time to roam around, find a place to eat calmly and then head to the concert.
Return that same evening to Florence from Montecatini offers only two options, one at 10pm and one at 10.50pm but which takes much longer.
View attachment 50
That's it, if you miss these two times, there are no other trains BUT there is a
last option with bus. It takes longer because it takes you first to Pisa and then you circle back to Florence, arriving in Florence around 2.37am but at least getting back.
View attachment 51
The easiest part of this entire trip will be Florence to Pisa airport - there are trains from SMN train station to Pisa Centrale every half an hour and from Pisa Centrale there is the Pisa Mover shuttle that takes you to the airport. So that part will be easy!
If you download the APPS for AT Bus and for Tranitalia on your phone, you can buy bus tickets and train tickets on the app, the most important thing is to remember to "activate" the tickets before you get on the bus or train because that's how you show you intend to actually use them.
If you buy tickets right before you get on buses or trains, it should be easy because it will give you the option right away. The apps also guide you to the bus stops so right now, with the technology we have, it has never been this easy to navigate and get to where you need to catch a bus!
Best of luck and let me know if I can offer any other assistance, and please let me know how everything goes after your trip, I honestly love getting feedback on your experiences because one thing is planning anf another is the actually execution!!