Thoughts on Itinerary - Rome and Cinque Terre

Looks good to me.....
While you are in the Forum area - go to the top of the Monumento a Vittorio Emanuele II - across the street from the the Forum . Also known as the wedding cake building. Take the glass elevator to the top and you have a great 360 degree view of Rome. You can stay up there as long as you like - the views are amazing. Nice way to orient yourselves to where things are located around Rome - in relation to the Forum. Happy Travels
PS: here is a link and don't forget your camera - great photo ops...
Vittorio Emanuele Monument

Rome- Vernazza-Florence-Rome.

After you leave Vernazza, stay the night in Florence, rather than going back to Rome and then make a day trip from Rome back to Florence.

For your last Saturday, consider a day trip to Ostia Antica vs Orvieto. Less travel and a truly fascinating enjoyable and low key activity.

After you leave Vernazza, stay the night in Florence, rather than
going back to Rome and then make a day trip from Rome back to
Florence.

That's is what I'd do too, and spend 2 nights in Florence. It feels more linear and less hectic to me than making two separate northernly trips involving backtracking (Rome>Vernazza> Rome, and Rome> Florence>Rome) versus just one (Rome>Vernazza>Florence> Rome). You'll also cut at least one train trip this way, and there's plenty in Florence to fill one full day +part of another. We've spent much more than that without running out of things to do, although it depends on your interests.

Day 5 (Tues) - enjoy Vernazza in early morning, leave midday on train to Florence (sleep Florence)
Day 6 (Wed) - Florence (sleep Florence)
Day 7 (Thurs) - morning train to Rome/open day (sleep Rome)

Sorry, but day trips from Rome to CT and Florence waste so much time on travel.

Rome deserves 5 days and Florence 3-4 days. I suggest staying in those cities.

The Hotel Balesteri in Florence has a great location on the Arno and is a very nice four star.

Thanks everyone for the good suggestions! I have been debating whether there was a more efficient way to work in Florence, so I will consider going there on the return from Cinque Terre. We don't want to cut out Cinque Terre altogether though as that is a new destination for me that the whole family is excited about and we are prioritizing that over Florence. I'm still tempted to stick with Florence as a day trip given that the high-speed train is only 3 hrs round trip, but I understand that overall that would be adding 3-4 hours on the train and would be backtracking geographically.

Any thoughts on whether hotels would generally let you leave luggage for 48 hours between stays at that same hotel?

Thanks for the expert advice everyone, and thanks for the Vittorio Emanuele tip, Frances, we will have to do that. Much appreciated!

I would include a stay in Florence, heading there directly on the afternoon of your arrival. This opens up more possibilities and would seem a more efficient use of your time.

Day 1 (Friday) - arrive Rome 9 am, lunch, then Freccia train to Florence - 3 nights
Day 2 (Sat) - Florence (David, Duomo, explore city)
Day 3 (Sun) - day to Siena, or Pisa, or Lucca
Day 4 (Mon) - train to La Spezia, then regional train to Vernazza - 2 nights
Day 5 (Tues) - explore CT
Day 6 (Wed) - enjoy Vernazza in early morning, leave midday on train back to Rome - 4 nights
Day 7 (Thurs) - Colosseum, Forum, Trevi Fountain, Spanish Steps
Day 8 (Fri) - Vatican, St. Peter's, Piazza Navona, Pantheon
Day 9 (Sat) - open day in Rome, or possible day trip to Orvieto or somewhere else
Day 10 (Sun) - depart Rome on 11 am flight

Thanks all - excellent suggestions all around. We will consider heading straight to Florence upon arrival or staying there on the way back from CT to Rome.