Ciao Lindsey,
With 3 kids, you definitely want to take it easy and remain flexible as far as what gets done. Also keep in mind you're coming mid-August - high season as far as the beach goes and many things are quiet/closed down on August 15 as it is a national holiday. If you're here then, plan to spend that day at the agriturismo.
Definitely spend time in the local area. In Chianti, you can see and enjoy gelatos and walking around the main square/sitting at a cafe while the kids run around and use up some energy (the main square is closed to traffic). Are you closest to Greve? There is a market there on Saturdays. All of the towns have playgrounds, not right in the main piazza but somewhere nearby.
You should also see Radda and Castellina in Chianti, as well as Panzano. Depending on the ages of your kids and their tolerance for a long day out, you could do a day in the Val d'Orcia (this is where Montepulciano and Pienza are located).
Another day in Volterra.
A day in Siena.
A half day in San Gimignano, the rest at the agriturismo.
A day in Pisa and Lucca (half a day in each).
Of course, you could also plan a day in Florence.
But whether you will be able to accomplish all this will only be seen once you're here. I don't think you will be able to do it all, because I highly recommend doing less to enjoy everything more. Don't pack too many things in one day and consider the driving times there and back. Pisa/Lucca are not far but putting them together in an all-day outing could be too much for them, so you really just want to have a general idea of the possibilities but see as you go what you end up doing each day.
Definitely do spend time at the agriturismo and relax by the pool - do the cooking lesson there (you can have
these two lady chefs come out to you) and wine tastings in the area.
I'd avoid the seaside just because it is over an hour drive to the coast and you need to get there by 8am if you don't want it to be too crowded. You can do it, but that means one day less of sightseeing elsewhere ;-).
Orvieto is too far, it isn't in Tuscany anymore so I'd leave that completely out (I deleted the new thread you started to keep your questions together - I had already answered yesterday
over here too, did you see it?)