What month are you travelling?
Hi....you barely have enough time to see Rome, Florence and "the coast of Italy", without adding Greece...Grab a guidebook, a yellow notepad, pencil and highlighter and have a good read...make notes of what interests you IN these destinations and then you will begin to see how much time you need for each one.
There is travel time in between each change in destination and it is always more time than the posted train schedule. Airports also will eat up a large portion of your day. Plan 1/2 day with every 'travel day' and sometime more in airports.
"The coast of Italy" is rather a long stretch...where exactly were you thinking? and when were you traveling?
Another aspect of planning is to think about your airports as you will most likely wish to fly in one airport and out a different one, in order to keep from backtracking and losing valuable vacation time.
IDEA of 12 days/12 nights, with travel time, optional day trips, 3 hotel changes, fly in Florence/out Rome:
--arrive Florence 1/2 day settle
--Florence 3.5 days/4 nights (day trip Siena, 2.5 days Florence)
--train to Naples, transport to AC destination 1/2 day
--AC destination 3.5 days/4 nights (day trip Capri, 2.5 days AC)
--transport to Rome 1/2 day
--Rome 3.5 days/4 nights
--fly out Rome