I would also add
PLEASE don't make this a see it all trip.
Build in some time for "staying" some place'
ASK your granddaughter.

ttt

Hi, as far as Italy is concerned, I would use one of the day in Florence to visit the countryside. Better with a rented car, but at least you can visit Siena with public buses. Lucca could be an option as well.
You can book your high speed train with the national railways http://www.trenitalia.com/ or with their private competitors (often cheaper) http://www.italotreno.it/IT/Pagine/default.aspx
If you would like to share my preferred places to visit in Italy you can have a look at my blog www.delightfullyitaly.com
Enjoy!

You need to figure the May holidays into this plan, too.

Personally, I think you will miss the part of "being there and enjoying it" because you will be too busy going from one destination the the next. I would pare it down to 3 or 4 places and stay put for several days at each place and just 'be there' and enjoy it.
Whatever you decide have a wonderful journey w/granddaughter.