Tuscany - Agriturismo

Given your areas of interest, I'd suggest an agriturismo near Volterra. Everything you've listed would be less than a two hour drive, even Radda and Montepulciano, which you didn't list but might want to consider. Siena is an hour away and Pisa is also just an hour's drive. Volterra is a lovely hill town. I would suggest you stay in the town at least until sunset, after the day trippers have left.

We were in the area in December and there were no crowds. Summer will be different. Parking may be problematic. I'd look into taking a bus into Siena from where you stay.

I did a quick Google search and there are plenty of agriturismos near Volterra. I selected one at random that looks pretty typical: http://www.agriturismo-volterra.com/uk/tuscany-villa-facilities.php

Your trip will be different with toddlers coming along. There's LOTS of walking (including steep paths) in these hill towns and it will be hot in the summer. You might want to consider that. I'd sure look for an agriturismo with enough amenities to keep the young ones entertained.

Thanks for the suggestion of staying near Volterra. I just checked out driving distances to some of the cities, and that looks doable. Now I can focus my search in that area over the next few months. I'm planning to go at the end of June to the beginning of July, but we should be okay in the heat and hilly walks. We just did three weeks in Lisbon and the Algarve with the kids and they did great. Temps were in the upper 90s every day! The hills of Lisbon were pretty steep at times, but we managed. The twins will be close to 3 by next summer, so theoretically it will be easier. With the Tuscany apartment, I mainly want a pool, although it would be extra awesome to have one with an extra kiddie pool and some farm animals too.

Glad my suggestion was of some use. It sounds like you've got some experience under your belt with the young'uns.

Prepare to fall in love!

Villa Arnilu near San Gimignano worked for us. Big pool, close to town and easy drive to Siena and Volterra and lots of other places. Easy train to Florence.

Driving from there to Pisa/Lucaa is a bit of a drive but doable. We spent a day at Pisa then at the beach in Viareggio. We did the beach because our 8 year olds (x3) and 6 year old needed a break. So, we went swimming in the Med!!! They loved it.

We did a trip all the way to 5 Terre but it was a long one. We took the train from La Spezia aftering driving there from San Gimignano. Long day but doable.

Just wanted to mention that you might have difficulties finding an agriturismo with AC. By definition, they are farmhouses used for vacation rentals. Very few have AC. However, traditionally they are built with thick stones and small windows. Keep the windows closed during the day and these apartments stay reasonable cool.

Volterra is fascinating, and from there you cana get to Luccca, Pisa and San Gimignano pretty easily, but then it might be better to move to someplace more central to the hill towns. Try looking at Castello di Montalto near Siena: it's right in the middle of Tuscany and easy to get to, so you can spend your time visiting where you want to go instead of on long winding roads that might make the children carsick. Plus, it's an agriturismo but also has the fascination of a real castle, not fancy though. There is a very nice pool and other entertainment for kids, and some of the apartments are actually separate old stone cottages with 2-3 bedrooms and private washing machines (useful). I don't know that you'll find airconditioning (only in some units) but the walls are thick, there are screens and shutters on the windows, and the night-time temperatures are cool even in midsummer. (Check the weather forecasts for Castelnuovo Berardenga). We love it!

We stayed at Agriturismo Marciano that was recommended on this site, and LOVED it! It is outside of Siena and a perfect spot to visit all the towns you have listed for day trips. The rooms were a nice size and the owner was wonderful. They also offer dinners for their clients and we had dinner there 2 nights, which was fabulous. They make their own wine and all the food was locally grown in their garden or neighbors. I think she offered a Rick Steves discount, I can't remember. I can't say enough about how much we loved it there.

Try this one.
http://www.tuscanyaccommodation.com/podere-le-cave.html

Close to all places you mentioned.

We stayed in Castellina in Chianti and it was a perfect location to get to Florence, Siena and Pisa by car.