Ideas & suggestions for areas between Rome & the Cinque Terre and Beyond

Orvieto Assisi Siena are on the way.
Not sure mean to want to cross the border into Nice.

Nice is quite a trek from the Cinque Terre, so I'd skip it on this trip.

Do you plan to have a rental car for part of this trip, or will you be using trains and buses?

Here are some possibilities, working vaguely from south to north. There are a lot of zigzags, and you'd have to pick and choose to create a coherent path.

Tarquinia (Etruscan stuff)
Orvieto
Assisi (other good spots in that area, like Todi, Gubbio, etc.)
Montepulciano
Cortona
Siena (south of Florence)
San Gimignano
Volterra
Lucca (near Pisa, north of Florence)

Bologna, which takes you farther NE than you might want to go, is interesting itself and also a good base for day trips to places like Ravenna and Ferrara.

With a car you'd probably want to spend a few days enjoying some of the small hilltowns in Tuscany--places that aren't so easy to visit if you're limited to trains and buses.

Think wine. Turin, Asti, and towns around there. Enjoy. Try a nice barolo for me.

We will look into all of the above. Thanks!

Nice and Genoa are both north of Cinque Terre so would not be on your route between CT and Rome.
Acraven's list is quite good and those places are on your way.

One thing I would add is that some of the locations are better with a rental car so if sticking with public transit your options are probably more limiting.

if you do have a car I recommend the Val D'Orcia area which is Montepulciano but also Pienza & Montalcino ; all 3 are very close together and great wine country.
Piece that base with additional stops or daytrips to Siena & Orvieto both a little over 1 hour away and you can have a great trip.

You could also use the same base and visit San Gim and Volterra quite easily.

Argentario is a beautiful location. There are beaches of various degrees from sandy to rocky and some beautiful views out over the sea. Tarquinia (Etruscan frescos and museum with incredible sarcophagi) is not far away. We combined both of these into a one-day stop in the area, although that felt somewhat rushed; I think 2 full days would have been better to allow more beach time. They're both on the main north-south train route along the coast. We took the train about an hour out of Rome to Civittavecchia, picked up a rental car and drove north from there. We were there in June and Argentario didn't feel overly crowded; I think summer weekend traffic can be intense, though, so you may want to plan around that. Since Argentario is almost an island it has limited access from the mainland.

I seriously considered Populonia but we didn't end up going due to time constraints. I like to balance city and countryside on our trips and the idea of a ramble in a historic, park-like setting on the sea was appealing:
http://www.turismo.intoscana.it/allthingstuscany/tuscanyarts/archaeological-park-baratti-populonia/

Lucca is a lovely town and you can train from there to CT change trains in La Spezia. It is a walled city with plenty of spa towns up in the hills. You can train to Florence from there also drive to Volterra or San Gimignamo, the resort town of Viareggio is just south of CT, Carrara in the mountains where marble is farmed, Pistoia to name a few. Pisa is really only a few hours stop.