Itinerary March 31- April 9, 2023

I always write it out day by day.
Friday Rome (arrival day can be less than productive, so keep that in mind)
Saturday Rome
Sunday Rome
Monday train to CT, hiking
Tuesday hiking
Wednesday hiking
Thursday train to Florence
Friday Florence
Saturday Florence
Sunday departure from Pisa--unless it's crack of dawn early, you can come from Florence. Otherwise, stay in Pisa or Lucca last night.
Thoughts:
Cut a day from CT? Nix Florence stay entirely and just day trip in? Do you want to stick to train and bus, or are you interested in renting a car trip for your rural Tuscany portion? What tours are you interested in?
I use Booking.com for accommodations.

You might want to look at Google Maps. The C/T is 280 miles from Rome, which is quite a distance. Pisa is an hour west of Florence by train and most travelers go through there going to the C/T. Most would catch Lucca on the way back--90 minutes by train from Florence.

I would visit Florence first and then take a train up to LaSpezia which is a short distance by local train from the C/T. On the return, Lucca would be a good stop on your way south.

David, it actually depends very much on what train you take--getting to CT can be just as fast/slow by train from Rome as from Florence. I often discourage people from going to CT from Florence at all (especially as a day trip), but as Keesus is still refining their itinerary, this is just a sample to give them some ideas. It may make better sense to make use of a car for part of the travel (dropping a car in La Spezia, for example). With this timeline, I would probably choose either CT and a bit of rural Tuscany or Florence and a bit of rural Tuscany, not both.

Here's how it could look coming the direction David suggested--my point is the Tuscan part is a bit compressed so some reworking may be in order, but it just depends on how much one likes to move.
Friday Rome (arrival day can be less than productive, so keep that in mind)
Saturday Rome
Sunday Rome
Monday train to Florence
Tuesday Florence
Wednesday Florence
Thursday train to CT
Friday CT
Saturday CT to Lucca or Pisa (depends on departure time)
Sunday departure from Pisa

We took the hi-speed train from Florence to Pisa and then onward to CT. Since we only had 6-7 hours in CT, we used the express train to move around the villages. The hiking path between the first two villages from LaSpezia was closed and I believe still is, yet I do believe these is an alternate less picturesque way to hike between them. The hiking paths and closures are shown on the CT website which is worth a look if you haven’t already. Have a great trip!

If I am reading this correctly, you are planning to be in Rome at the start of Holy Week, the most important time in the calendar of the Catholic church. And you don’t have lodging reservations? Yikes! Get that pinned down asap. Not only will Rome be full of pilgrims, but your time frame coincides with spring break for the schools in many countries.

The standard tickets to the Vatican Museums (including the Sistine Chapel) are sold out for the entire length of your trip. There are some early-entry and tour options available on the Vatican Museums website for the day you hope to visit (Saturday, April 1). I would expect those to sell out--perhaps very soon. I'd suggest you get something booked immediately if you're confident the trip will take place and seeing the museums is important to you. (I don't think the tour tickets are refundable if something goes wrong.) You don't need a ticket for St. Peter's basilica itself, though you may encounter quite a line. Some tours of the museums allow you to cut straight through to the basilica at the end of the tour, but that presumes you are satisfied with what you've seen of what is on display in the huge museums at that point.

There are also commercial companies offering tours of the Vatican Museums and/or St. Peters. They will generally cost you more than similar options offered by the Vatican itself, but their offerings may not sell out quite as fast.