Hey! I stayed in Montecatini Terme last April, it's a great little town to base your trip out of. Not much to do or see, in town but it has all the amenities you would need, and some ok food. Make sure you take the funicular up the hill to Montecatini Alto, much more like Italy and less like a tourist hotel zone. Great food, small but very cool.
We were able to use the train station (there are two in town) to go to Florence but it can connect you just about anywhere. We drove mostly, easy access to everything in Tuscany.
We did, though it's not for the faint of heart the train to Rome from MT in one day! We are young(ish, I'm 37) and aggressive travelers, so we took the 7am which had us into Florence, then connecting via the "fast" train to Rome by 10:30am. We took the last train out of Rome to Florence and arrive back in MT the same night around midnight. Walked about 10 miles in Rome that day! Again, this is a lot. I'm glad we did it, but not for everyone.
We visited San Gimignano, Siena, and Volterra from this spot quite easily. Lucca and Pisa are nearby, and one day we drove to La Spezia to catch the train to Cinque Terre for a nice day trip!
Sounds like you likely got the same deal via Gate1 perhaps that we did (also first time to Italy), the drive from Milan to MT is beautiful. You will have a great time. If I had to do it all again, I would spend more time in Florence and the Hill towns of Tuscany, skip Rome for another time.
Cheers!