Buongiorno,
I believe that your best base would be in Florence or just on the outskirts of Florence. Every place you mention below can be reached by public transport from Florence with no bus/train changes along the way and with fairly frequent and reliable transport.
Lucca - train
Siena - bus
Cinque Terre - train
Montecatini - train
Arezzo - train (this itinerary makes a circle around the city and you can start at the train station)
Chianti - bus (any one of these articles about the towns also gives you detailed info on how to get there with public transport)
Buon Viaggio,
Donna Denise
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