Ciao Diana,
Your itinerary sounds very good for 7 days in Tuscany! Moving around by public transportation isn't always the easiest in Tuscany but it really depends on where you're going - it can be done with some patience and planning!
In your itinerary, the hardest to get to without your own car is Volterra but not impossible. Usually I'd recommend going from Pisa to Siena, and then getting to Volterra by bus but it isn't impossible to do in the order you want to do it.
First, you can get to Lucca from Pisa by train very easily. There are also buses but I always suggest using train as the first option. You can check time tables for trains on
www.trenitalia.com.
As for Volterra, that is the more tricky part - coming from Pisa, you can actually also try the train. You have to first go to Cecina, and then head on to "Volterra-Saline-Pomarance". Check time tables at the link above.
By bus, instead you you have to go Pontedera and from there catch another bus to Volterra. You can check buses to Pontedera and to Volterra here:
http://www.cpt.pisa.it/orare/extraXlocalita.htm
From Volterra, you can take a bus to Colle Val d'Elsa and from there change bus to head on Siena. You'll find schedule to Colle on the link above, while from Colle to Siena you'll find the schedule here:
http://www.busfox.com/timetable/
All you need to know now is the times that are available, so that you'll know once you're here which trains or buses you want/need to take.
You get tickets once you're here, right before traveling when you're at the station.
Day trip from Siena to Florence is really doable by bus as well (timetable at the busfox site above) but Chianti will be a little more difficult to do in general, as it is an entire area. You COULD do a guided tour from Siena to Chianti, such as this one:
http://www.partner.viator.com/en/83...eve-in-Chianti-from-Siena/d944-5493FLORENTINE
You could also do a day trip to Montepulciano and Pienza
Check out more tours here:
http://www.partner.viator.com/en/8357/travel-tips/Tours-in-Siena-and-from-Siena/ttd
A private car with driver as suggested by john is always another option, you just have to check out his rates and see whether they are within your budget and something you want to try!