My wife and I will be in Rome from 03/11/2019-03/15/2019, Florence 03/15/2019-03/18/2019 and back to Rome from the 03/18/2019-03/21/2019. This our first trip to Italy and my wife's first trip to Europe. So far I have booked a guided tour of the Vatican for 03/12, and a guided tour of the Colosseum for 03/13. I bought the ticket that includes the Forum and Palatine Hill. I was thinking we could go to the Forum that same day or the next. Our hotel is located walking distance to the Trevi Fountain and the Spanish Steps. We will travel to Florence on Friday the 15th and return on the 18th. Other than the Doumo and David, and the Ponte Vechhio bridge, what should we see? Once we return to Rome I was thinking of going to Tivolo for a day trip. We fly back on the 21st. What other sites or experiences should try? Its a leisurely trip for us, or we plan it to be.
Check out the Bargello in Florence, and Piazza Signoria, and Palazzo Vechio, and Vasari Corridor. Of course, the huge art museum called the Uffizi.
It's Tivoli, not Tivolo. Great little town. You could see Hadrian's Villa on the way there. Another consideration might be Ostia Antica, the Roman port from a couple of thousand years ago.
Rick has a star-rated sightseeing list and a number of audio tour links at https://www.ricksteves.com/europe/italy/rome, and similar for Florence, but I'd also want the printed guidebook to go along with those.
Unless your arrival on the 11th is late in the day, I am puzzled about splitting your time in Rome.
Going to Florence on your arrival day and doing all your Rome days at the end, would have cut out a hotel change and left more time for touring.
Happy travels
We greatly enjoyed the Museo Galileo while in Florence.
Whatever you do, be prepared for some cold weather. When touring, that means layers & portability. Rome--at least from my experience--is surprisingly warm (40's-60's) but twice I've been in Florence in late winter/early spring when the wind howls and it's just plain cold. Zippered fleeces work well, but my ace in the hole was a lightweight down jacket--the kind campers use--that when not in use folds up into a little ball into a backpack. I have a Marmot brand.