Some things to think about.
Italy is usually hot in June. The cities you have picked will be teeming with crowds so some stress trying to keep track of little ones.
If you stay near Termini think about how you will get to sights. Bus or metro? Walking is usually not feasible, too far and hot. Frankly Rome is a challenging destination to get around in even without young children. Scooters at intersections pose a hazard also.
Florence is mostly about history and art what will the children do?
Make sure that you are not setting yourself up to be frustrated by having things you have always wanted to see not work out because the lines and crowds don't work out for the children.
There's a huge advantage to renting a car with children because each child has his own space and the driving time is relaxing because you know where everyone is and that they are safe and can't run into traffic or onto train tracks, etc.
You can always bring car seats with you for rental cars then abandon them when leaving.
Suggest flying into Paris and immediately rent a car, see places, return car and see Paris.
