Itinerary

Lucky you! You've got the time, so you just need to narrow down the details.
Are you open to driving in Tuscany? That will be a deciding factor, but I definitely would not want to do multiple one-night stays.

We have driven in Spain, Australia, Mexico but hesitate to drive in Italy because we have read horror stories about people receiving fines long after their trip--exorbitant fines for minor infractions that they were not even aware of. So--if going by rail, would you suggest several day trips from Florence or less time in Florence but adding on a few nights in a smaller town in Tuscany?

If my husband and I can do it, seriously anyone can, but I understand. Splitting time between Florence and a small town would work nicely. Siena is mentioned as a place with relatively decent public transport connections. If you stick with Cinque Terre, it is easy to reach from Lucca, so first I would spend a lot of time with a Tuscany guidebook to narrow down your choices. Once you know what exactly you want to see, it will be easy to determine the best base.

The train trip from Barcelona to Milan is almost 14 hours and almost 400 miles. Have you considered flying on Vueling Airlines, a budget carrier?
You could take a day trip up to Lake Como--around 45 miles one way from Milan.
Florence is a great place to stay and do day trips from. Siena is a great day trip by bus which takes you into the city center. You could also rent a car for a day and easily visit towns like San Gimignano, Certaldo and Volterra. I wouldn't suggest driving thru Tuscany and staying in different towns because the region is just not that large.
The Cinque Terre is a little far from Florence for a day trip, and it's more of a June-September travel season.
Naples and Sorrento are also pretty far south of Rome--150-200 miles. And since you've already been to the Amalfi Coast, your time might be better spent in Rome. That's a city I don't get tired of.

I looked up the train trip from Barcelona to Milan on Seat61 which makes it sound quite appealing with an overnight stay in Marseilles.

Not sure how long the train ride from Florence to Sorrento would be. We really do want to stay in Naples/Sorrento area and would like to visit Pompeii and maybe Capri if possible. Would a low cost European air carrier be a better option?

Not sure how long the train ride from Florence to Sorrento would be.
We really do want to stay in Naples/Sorrento area and would like to
visit Pompeii and maybe Capri if possible. Would a low cost European
air carrier be a better option?

You can look up the train time to Naples on the Trenitalia website. From Tuscany, flying would probably not save you any time. From Naples to Sorrento (if you prefer staying there), you can take the Circumvesuviana commuter train.
Provided you have ample time, I don't see it as a problem to go south of Rome and then back. Those are just the realities of air travel and geography. I would at least check Naples flights though. Naples to Rome is a short train ride, but if you decide not to go back to Rome, there is a direct airport bus called the Fiumicino Express that I would look into for convenience.
I would consider staying on Capri if you find it compelling enough to go -- there are downsides to day trips. See the recent NYT article about Capri. It sounds much nicer if you can give it time to soak up leisurely.

While Tuscany may not be that large, it certainly seems like it is because the roads are windy and there are thousands of hill towns crammed into that small area. I have only focused on the northern and southern corners.

Three/Four ideas:

1) Consider a northern "Lake Route Circle" Maggiore, Lugano, and Como (and even perhaps Lake Garda) and back to Milan over five days. Best to stay in one place and travel about by train, bus or lake boat. The scenery is great, the food is great, but the weather may be variable in the southern foothills of the Alps.

2) Stay at an Agrotourismo in Tuscany. We loved ours for the five days we spent there one autumn. They range from rustic (cook your own meals in an old farmhouse on a hilltop - but with the farm store selling local boar, cheese, etc.) to more full service places with meals, cooking classes and arranged tours (wineries?). We also splurged on a truffle hunt, and since the truffle hunting dogs found two there was the option buy them directly from the hunters at the end (it is a small business niche of great local interest, but not cheap). Just google "Agritourismo, Tuscany."

Our experience with driving in Tuscany was that it was scenic and once away from the Autostrada... really basic two lane roads in the hills S of Florence..... Great fun getting to San Gimignano and Volterra, but only for the passenger. Day tripping or overnighting in Siena are both do-able. There are lots of wineries that are reachable by car, but beware the tasting and driving.

3a) When we went to the Cinque Terre in April it was cool and rainy on a few days, but not too many tourists like us. If I was going back soon I would try to stay in Corneglia (the one upon the hill with the long walk up). Plan for three days plus half a day getting there and getting back to Florence (3.5hours Firenze to Corniglia).
https://www.lefrecce.it/en

3b) I love Pompeii, Herculaneum (Ercolano) and the Naples Archeological museum (although it is still in the 19th century as to exhibit designs). Many guidebooks recommend taking the Circumvesuviana train to the west end of the site, which does make sense if you are doing a day trip from Naples. But you can take a regular train to Pompeii from Naples - it just takes you to the town of Pompeii from which you can enter the site from the east end..... Staying in Pompeii is local Italy which we loved - real pizzerias, and ladies in the laundromat who all do the clothes handing while you lunch and wander. (4 hours Florence to Pompeii). You can then day trip up the volcano, go to the Museum in Naples, go to Herculaneum, etc.

We have driven twice in Italy and have had one minor traffic fine... not too big and not something to get in a knot about given the other costs involved...kind of like a speed camera fine in any American town.

You have the time to make it a really worthwhile trip.Have fun.

Bologna is a very good base for day trips by train . . . . Padua, Modena, Ferrara, Parma, even Ravenna (though much better as an overnight). Florence is a good base for Siena (bus), Pisa/Lucca (train). It's a long trip to the Cinque Terre, so that's much better as an overnight.

I've stayed in Naples and Sorrento and liked them both, though they are about as alike as apples and bicycles. There's more to see in Naples but Sorrento is more picturesque (and more expensive). To get to Sorrento from the north, you'd take a high-speed train to Naples, then the local Circumvesuviana (really a metro, not a "real" train) to Sorrento. Then the same route to Rome.

With the days that you have, your need to cut down your itinerary.

I would not do a day trip to Cinque Terre from Florence. If you are in Milan, why not go to CT first.

For Tuscany, consider Sienna or Lucca.

Naples, Sorrento and the area require more days than you have available with your other plans. Consider skipping this until another trip.

I have been to Bologna, it was OK, but I was not awed by the city and its unfinished cathedral. Loved Ravenna.