rickg523

Welcome aboard!
What you basically have is 4 trips in 16 days.
3 nights in Florence - good

2 nights in CT - I'd stay 3, but 2 is fine. I'd also find (basic) accommodation in Vernazza (or you could try Monterosso if basic doesn't suit you) to get the full benefit of your visit. Don't stay in La Spezia. Here a tip. This website lists the cruise ship schedules for La Spezia. Try to pick a time for your trip when the smallest cruises are in Port, because all those people are going to CT. And it can get ridiculous if you're there on the wrong day. This is also a good reason to stay in CT. The cruisers all head back to their boats in the afternoon, so you'll have the evenings minus the mobs.
http://www.cruisemapper.com/ports/la...17-06#schedule
Its set to June, but find your own dates, then click on the ships in Port to see how many passengers they will be dropping on poor Cinque Terre that day.

5 nights in Rome - you can train from CT to Rome (cxn La Spezia) in about 4 hours. Consider the Pantheon the center of Rome and stay as close to it as you can. Some may say to stay a day or two less in Rome with more time elsewhere. And they may have a point. But I can't say that. I find Rome endlessly fascinating and could personally spend the entire 16 days there. To me, it's among the top three best cities on earth. You can easily do a day trip to Orvieto from Rome. You can buy train tickets the day you go. If you can manage it - makes for a long day - try to visit Civita di Bagnoreggio while you're there. Its a bus ride between the two towns.

4 nights in Campania - here I suggest using Sorrento as your base. Its touristic, but still charming. There are good accommodations of all types available. Most importantly, it's the easiest transit center for visiting Capri, Pompeii, the Amalfi Coast and Naples. Transit options include boats to Capri, Positano, Amalfi, and Naples. Trains to Pompeii and Naples. Buses (a thrilling ride if you sit near the front on the right side) to the Amalfi Coast. From the center of Sorrento, the train/bus depot is 5 minutes walk. The port maybe a few minutes more if you use the Lift near the Hotel Tramontana. Each of your destinations in Campania is a day trip. So you could do them all, but maybe pick three and relax a little.

I may be counting you an extra day, depending on when you arrive and depart, but you can do this trip.
As a point of reference, I planned a 3 generation family trip (6 people) to Rome, Campania, the Lakes, and Venice over 21 days. None but I had been before. I would consider these kinds of trips an orientation to Italy.

Now that you've found this forum, I highly recommend you pore through the posts here. They are a wealth of information. And the experts here are awesome and beyond helpful. As I said, I've been to Italy a lot, but I'm checking this forum all the time and learning new things regularly. My experience pales compared to some here.
And as you build your itinerary, check back for specific tips and recommendations.
Have Fun.