If you're visiting your niece then stay in Florence. You can easily do day trips to Lucca, Pisa, Sienna, Rome.

You will need a car for small towns in Tuscany or for staying in a villa outside of a town.

I would fly into Florence (or Pisa) and fly home from Rome. Or the reverse.

With 10 days, you could stay a few days in Florence, a few days in Rome and a few days somewhere in between. If you have a car, the in between stay can be essentially anywhere in Tuscany. Without a car, you'd need to pick a town (not a remote-ish villa) with good bus and train connections to the other places you want to see. Siena, San Gimignano, Lucca, Pisa and, to some extent, parts of Chianti are easy day trips from Florence using only public trans.

The hard part is deciding/limiting what you want to see/do because the list of options is very long. Resist the temptation to go to every "must visit" place.

Sounds like a good trip. I would do at least 3 full days in Rome, 2 is not doing it justice at all. I would choose 1 location in Tuscany (Florence or Siena area) for 6 nights and do day trips, 1 in Rome for 4 nights.

If you really want to see the coast it is do-able but you would have to steal some time from Tuscany.

As mentioned above by Jean, if you can, fly into Rome and out of Pisa, or vice versa.

I would not do a day trip to Rome as this is simply too short a time to even begin to explore the city. I would also suggest flying into Florence or Pisa and leaving from Rome after spending 3 full days/nights there at the end of the trip.

I would base in Florence and do day trips to Siena, Chianti, unless you are interested in renting a car and exploring the countryside on your own.

ditto, kf