Eric,
You can take fast trains directly from Rome to Florence. Your fastest route for Florence-Nice has one connection in Milan. You can easily buy those tickets after you arrive in Rome.
For Nice-Paris your choices are daytime TGVs--some direct and some with connections--and a direct night train. Since August is the month when "everyone in France is on vacation," the cheapest fares are long gone. However, if you book ASAP, you may still be able to get a fare for a little less than full price. Your best fare will probably be on the iDTGV.
Go to www.tgv-europe.com. To keep the site in English and to keep from being bumped to Rail Europe which rarely offers discount fares, choose Great Britain as both your country of residence and the country in which you will retrieve your ticket. You will be able to pick up your ticket in Nice even though you originally chose Great Britain.
The Cote d'Azur will be packed in August. I suggest you book a hotel ASAP.