rome to florence/siena/cinque terre

First question are you flying round trip in and out of Rome or open jaw?

Hi Jo!
Nope, we're not returning to Rome :)

Your plans don't make much sense because you back tracking to places a lot. Your most logical itinerary is Rome > Florence > Cinque Terre > Provence. Here is what you should do.
Arrive Rome, visit Rome.
Train to Florence. If you don't plan to have a car, you should stay in Florence and use it as your base to visit the Tuscan towns you wish, including Siena. Siena is an easy day trip from Florence by frequent bus (75 min). Siena, like Florence, is also verboten to cars, therefore visiting by bus works well. You can visit Siena's sights in less than a day. From Florence you can also visit by bus or train many other famous Tuscan towns, such as San Gimignano, Volterra, Certaldo (Boccaccio's birthplace), Cortona, Arezzo, Lucca and Pisa, and countless others.
After you are done with Florence and Tuscany, you should head to the Cinque Terre by train. Along the way it's a perfect opportunity to visit Pisa's leaning tower, since you will need to change trains in Pisa (leave your luggage at the station's baggage depot, while you visit). After an hour or so in Pisa, proceed to La Spezia by train and from there to any of the 5 Terre. It is safe to arrive any time you want in the evening. You are in the Cinque Terre, not in the South Bronx. The worst it can happen at night is to stumble in a stair step on the way to your hotel. Heavy luggage is not advisable at the CT because the towns are steep and from the station you will likely have to walk uphill and walk steps too. I recommend staying in Monterosso because it's bigger (more hotels and amenities), has a nice beach, and the walk from the station to the village is not a steep uphill like the other towns.
Lastly, from the 5 Terre you will proceed by train to Provence. That one is a long trip, with likely train changes in Genoa, Ventimiglia, Nice and Marseille. There is a train option from Monterosso to Aix-En-Provence TGV station, that requires one change only (in Ventimiglia), but you need to depart from Monterosso at 4:55am. Plan on 10 hours on trains that day.

Hi Roberto!

Wow thank you so much for the info!

Would it make more sense to cut the journey and have an overnight in Nice?

Also, what do you think if we reverse the itinerary: Lucerne -> Florence/Cinque -> ROme -> fly to Avignon

If you want to spend an overnight in Nice is up to you. Certainly possible.
I'm confused about your reversal of the trip. Lucerne is in CH, and that wasn't in your itinerary before.
Avignon has a ridiculously small airport and there are no flights from Rome. To fly from Rome your only options are MRS or NCE.

Roberto's right on the money about your itinerary.

Now when it comes to luggage, you should seriously limit your wardrobe. On our family trips, we carry one 22" rolling bag and a small backpack or small handbag. And I don't carry luggage for anybody. It's simply torturous to haul a couple of big suitcases anywhere. On my recent European trip, I didn't even wear 1/2 the clothes in my rolling bag. Experienced travelers can travel indefinitely our way.

Please don't pack more luggage than you can handle yourself in a variety of situations - train stations, hotels, and narrow cobbled streets, all of which often include stairs. "Huge luggage" is asking for frustration. See tips at http://www.ricksteves.com/travel-tips/packing-light. If you decide to leave an extra bag in storage at La Spezia train station, you can do so for a few euros per day. They will want your bag to be locked and will take a photocopy of your passport. You may have to ring and wait for an attendant to arrive at the window.

There's no problem with arriving in any town in the evening, but do let the hotel know the time you plan to arrive.