wine tastings and spas..... a car would be useful!
Siena is a beautiful town, quite different from Florence so you'll get to see different sides to Tuscany on your upcoming trip.
Definitely visit the inside of the
cathedral - beautiful mosaic pavements! - and climb up the
Torre del Mangia to get a beautiful perspective over Siena.
Now getting to the surroundings around Siena without your own means of transportation will make things a bit more difficult.
If you want to do it on your own, there is
this website where you can check bus schedules around the province but at the moment it is down and I can't get verification for any of the places I was going to suggest... Let's see if it comes back up soon, as it is definitely needed to see whether you can get to the following places with public transport or not.
For
spas, I am thinking you intend the hot springs that are pretty common across Tuscany? Close to Siena, there are two places I'd recommend:
-
Bagno Vignoni - a bit beyond San Quirico d'Orcia - to access the hot springs you will need to enter into one of the spa centers
-
Terme di Petriolo - this has a spa center as well as an outdoor area right by the road and bridge that allows you free access to the hot springs
There are two other places of note:
Rapolano Terme and
Bagni San Filippo.
In Rapolano Terme there are two thermal centers:
Le Terme di San Giovanni and
Complesso termale dell'Antica Querciolaia
while in
Bagni San Filippo there is a thermal center called "Nuove Terme San Filippo" as well as as an outdoor area accessible to all but you need to head a bit into the woods outside of the town - there is a hole called Fosso dell'Acqua Bianca as well as a pool full of water and mud turned completely white called the "white whale"

I've actually never been there but heard about it and think that might be an interesting experience.
Wine tastings: from Siena you can either head into
Chianti as well as south into the Valdorcia for the great Brunello wines of
Montalcino or the reds of
Montepulciano. You should
contact Vinarium, a tour company specialized in wine tours as they offer tours in both areas and see what most appeals to you.
Tours: A
tour of the Chianti area with wine tastings that we would recommend leaves from Florence, not from Siena - would it be possible for you to organize it while you are in Florence?
Hope these suggestions are useful and that you have a great week in Tuscany!
P.S. The Duomo in Siena has introduced the
OPA Si Pass which allows you to visit the cathedral, crypt, museum, baptistery and another museum for just 10 euros - a great bargarn and highly recommend it! You can book it ahead of time and just pick it up at a very short line or buy it at the ticket office directly.