There is one Nice-Rome route that hugs the coast all the way rather than going inland. Take a regional train from Nice to Ventimiglia, departing at 05:23 and arriving at 06:12. Take an IC train from Ventimiglia to Rome, departing at 06:33 and arriving at 14:00. Or depart Nice at 10:23, connect in Ventimiglia and Genova, and arrive in Rome at 21:57.
For complete timetables, go to www.bahn.de and click on the drop-down flag menu for English. After you get summary timetables for Nice-Rome, click on the arrow to the left of each departure time to see the connecting points. Then click on "Show intermediate stops" or "Show map."
Do the same for Nice-Firenze (Florence).