Public transport is certainly the better way to travel on a tight budget since you'll save on the rental, gas, toll-roads and parking. Trains in Tuscany are the best way to travel between the larger towns since they offer more frequent connections, with buses providing service to many of the smaller towns in between.
If you are thinking of depending solely on public transport, I'd recommend staying either in Siena or one of the larger towns in Chianti (Greve, Gaiole, Radda, Castellina) or to the south of Siena in Montepulciano or Montalcino so that it . Offerings within the city of Siena are easier to find, those around Montalcino and Montepulciano are often outside the historical center unless you go for a hotel, bed and breakfast or rooms.
Probably Siena is the most practical since you can catch buses to most everywhere you'll want to visit or a train. Some day trips perfect from Siena include Volterra, San Gimignano, Gaiole and Greve in Chianti to the north, Montepulciano and Montalcino to the south.
A whole week in Cinque Terre seems like a lot - are you planning on doing some hiking in between the villages? Trains connect all of them so that is the easiest way to visit all of them, but hiking between them also means paying a daily pass to use the main #2 trail. The Cinque Terre is a park after all, but plan what you want to do so that you buy the best pass (they offer multi-day passes with or without train (unlimited use if included), so costs will vary). I was recently there for 2 days and we saw all of them - I wonder if you might prefer to spend a little less time there and spend a few more days in Tuscany...
You can check train schedules:
www.trenitalia.com
and bus schedules in the Chianti area closest to Florence:
http://www.acvbus.it/index.php?SEZ=5
For buses from Siena, check this new tool:
http://www.busfox.com/timetable/ Seems to work pretty well, telling you the bus lines and cost for traveling all around Siena. Here is a map of the lines:
http://www.sienamobilita.it/mappe/Mappa_EXT.pdf
Hope you have a wonderful time in Italy!

Post any more questions you might have, we'll try to help as best we can - there is a lot to see in Tuscany but you'll have to decide for yourself what you want to see on this trip, maybe get a taste, and think that you can always come back for more.