Chianti is very central and perfect for day trips and wine
Ciao Jessica and congrats on your upcoming wedding!
I agree that if you've already been to the major cities then you definitely need to see and experience the countryside and the smaller hilltop towns in Tuscany!
In this case, a car IS necessary because as soon as you explore the smaller towns, bus service is not great and train tracks do not cross through the beautiful vineyards of Chianti or areas that you'll want to see. Plus, the region is made up of hills and hilltop towns. Even with a train stop nearby, the stop won't be in the hilltop town but at the bottom of it... and if you stay outside of the town, do you really want to trek mornings and nights along the roads to get into town? I wouldn't generally recommend it - roads are narrow with no area for pedestrians, a car is just safer and more convenient.
A whole week is perfect! With a car you can stay in the countryside where the villas and farmhouses are located and these are generally rented by the week. What month will you be here? Peak season is from June-August so if you're coming then, I'd suggest you start looking at possible places you like to book early.
As for places to visit.... there are many areas off the beaten track but Tuscany is pretty well known so there are visitors everwhere. And keep in mind that even in Florence, with all of its tourists, as soon as you leave the main area around the main sights, you find genuine, local trattorie and restaurants where the locals eat out at.
As for a central location, the first place that comes to mind is Chianti but this is a very wide area in the center of the region.... from San Gimignano to Arezzo, and from Florence to areas around Siena. So you have a lot of choices to choose from.
I'd recommend visiting San Gimignano, the hilltop towns in Chianti (Panzano, Radda, Castellina), definitely Siena if you haven't visited it yet and the area to the south of Siena called Val d'Orcia. And Arezzo - it is definitely off much visitors' lists but it is a charming town with many treasures, including the frescoes in the San Francesco church.
I suggest you look at the following, all in great positions from which to visit the places mentioned above:
Podere Cunina -
http://www.tuscanyaccommodation.com/podere-cunina.html
Romitorio di Serelle -
http://www.tuscanyaccommodation.com/romitorio-di-serelle.html
Il Cellese -
http://www.tuscanyaccommodation.com/il-cellese.html
Fattoria Viticcio -
http://www.tuscanyaccommodation.com/fattoria-viticcio.html
San Giovanni in Poggio-
http://www.tuscanyaccommodation.com/san-giovanni-in-poggio.html
If you don't want to have your own place, Romitorio and Cellese both have restaurants.... and this B&B is really romantic and in a hilltop village so there are pizzerias and restaurants just steps away: Palazzo Malaspina
http://www.tuscanyaccommodation.com/palazzo-malaspina.html
Let me know what you think so I can offer other suggestions if these don't match with what you had in mind!