It would help to know where you've already been in Italy besides Florence. Also, will you have a car anywhere or are these trips by public trans only?

We also chose La Spezia as our base and did day trips to CT and Portofino however,as Pisa train station has a luggage keeping facility,we did it on the day we left our hotel,while going to Florence-which saved time.
From Florence, you may include a day trip to Siena and San Gimignano.
From Naples,you may have day trips to Capri+Anacapri,Ischia and Procida islands and Sorrento.
Have a nice time in Italy

Dear Jean: Thank you for the request for more details. I lived in Milano as a child and did a lot of family travel..was Stationed in Sigonella, Sicily, 2yrs and went to Naples several times (1980s) Since then we have been to Argengo, Lake Como, Venice, Florence, Siena, Assisi, Milano, Bergamo...

Our frustration with our last trip was we felt we didn't get a local experience...Can't avoid it in Florence (Rome as well) because of all the tourists..

Our intent is to use trains....We can rent a car if necessary for day trips (more if cost effective)...had a terrible time returning the car in Florence after a day rental going to Assisi and Siena (return was on a holiday...couldn't find the return lot...what a nightmare!)

I guess I should add that while my husband wants to see museums...I am a professional photographer and want to shoot people and locations....I particularly would like to visit places that are off the beaten track so that my images don't look like every other photographer/tourist. Are there any festivals that would be fun to check out?

Personally, unless I was spending some nights away from all of these cities, I wouldn't rent a car. Sadly, that means missing what I would call off the beaten track. Anyway...

From La Spezia, you could bus to Portovenere or train to Lucca.

From Florence, Arezzo and Montecatini Terme (plus the Alto town) are easy by train. San Gimignano is doable by buses or combo of train/bus but a little more involved. I couldn't spend an entire day in San Gimignano myself, but others enjoy the town. I'm a little surprised you plan to see Parma and Ravenna from Florence because the train rides are close to or greater than 2 hours each way (beyond my personal limit for day trips).

Perugia has great bus service to other parts of Umbria. We especially enjoyed Gubbio and Todi. Some trains to/from Spoleto take about an hour. Tiny Spello is about halfway between Perugia and Spoleto.

You've mentioned the usual excursions from Naples. I'll add the island of Ischia, the Greek ruins at Paestum and the royal palace at Caserta.

There are loads of options from Rome. Tivoli, Hadrian's Villa, Orvieto, Bracciano (Lake Bracciano), Castel Gandolfo (Lake Albano), Viterbo.