Regarding Pisa, this is easily doable from Florence. However, I found Pisa to be the least enjoyable part of our two week trip to Italy. If you really want to go, go as early in the day as you can. Once the hoards of tour-bus tourists hit, the place can be miserable. Be prepared to be yelled at (hey, you're ruining my picture!), pushed around (you're blocking my picture), and smacked with selfie sticks (oops, sorry!).
My advise, take a really early train (regional to S. Rossore), see the tower and cathedral, then go spend a nice late morning and afternoon in Lucca.
With 15 days to play with, in my humble opinion, I would catch a train directly to Venice and work my way back to Rome. (But, keep in mind that it's going to be hot in July.) The train will put you all to sleep so I would set your watches to periodically wake you up. The good news is the train ends in Venice so you won't miss your stop.
Verona is easily down via train from Venice as a day trip. Padua and Vicenza are also good candidates for day trips from Venice.
Florence is 2 1/2 hours from Venice by train. Florence makes a really good central hub for exploring Tuscany. But, if you want to drive, choose a hotel on the outskirts and don't even think about driving into the historic center. (Google ZTLs or Italian restricted traffic zones.)
If you want to go to the Cinque Terre, I would do it between Venice and Florence but getting in and out on trains can be time consuming due to its remoteness.
Sorrento is also a little out of the way if you are wanting to just pop in and pop back out. If your intent is to visit the Amalfi, you might want to consider a couple days dedicated to just the Amalfi and pick Salerno or the city of Amalfi. If your intent is to visit Pompeii, you might want to consider an escorted day trip from Rome.
Finally, I prefer to pick central sightseeing hubs and mix in-situ sightseeing with day trips. This minimizes unpacking, packing and transportation time. I have rented apartments in Venice, Florence and Rome and feel it's the only way to travel. You can get a really nice 2 bedroom/2 bath apartment for the same or less than multiple hotel rooms. Plus, in Europe, many "double" beds convert to two single beds.
Good luck.