Hang out in Cortona
Take the time to get to know the town, find its stores and amenities as described here.
This itinerary is awfully calm - we don't suggest that you do too much each day, but rather frequently return home to Cortona to sit on the wall, do a sketch or just enjoy the view.
Siena has plenty to offer for a day's visit. Start in Piazza del Duomo with a visit to the Duomo famous for its intarsiated marble floor and for the Piccolomini Library with renaissance frescoes by Pinturicchio.
Across from the Duomo is the large museum complex of Santa Maria della Scala, previously a pilgrims' hospice and now used for special exhibits.
Thermal baths and outlet shopping
The outlet at Valdichiana has a good selection of mens and womens clothing and accessories, as well as kitchen stuff stores. Definately worth spending a few hours (and euros) here.
As for thermal baths this area is rich in sulfur waters, so you can choose from Bagno Vignone, Rapolano Terme or Chianciano Terme. The water is the same and wonderfully warm in all cases; what changes is the level of luxury at each thermal bath establishment (and also the price changes). Bring your flip flops, bathing cap, and a robe!
Wine & cheese party and Crete Senesi
Get the best of Tuscany's food with cheese in Pienza, followed by wine tasting in nearby Montepulciano. Pienza is also historically important - it is a town entirely planned in the Renaissace, with an interesting church and other public buildings. Just outside Montepulciano don't miss the Church of San Biagio.
Arezzo and Monterchi
Arezzo is the closest large town to Cortona. There are paid public lots and it's easy enough to find a spot to leave your car. Get a map and explore this city, starting perhaps with the church of San Francesco and its frescoes of the True Cross by Piero della Francesca. There is another figure by that artist in the Duomo. If you're lucky enough to time your visit with the Arezzo Antique Fair that happens the FIRST sunday of the month, be prepared for crowds but also for the possibility of finding a real treasure to take home!
After leaving Arezzo, if you liked Piero della Francesca, you can take a small road to Monterchi to visit the Madonna del Parto, a fresco by Piero that is believed to have magical properties with regards to assisting in pregnancy and birth. Local women still pray to her.
More small towns or a hike
If you step into the bookstore in Cortona you can buy a hiking map of the area, or just follow any red and white striped signs you see leading out of town. A long hilly hike followed by a nap in the park is just what you need to complete your trip.
If hiking is not your thing, go check out any of the other small towns in the area, including San Giovanni d'Asso and its truffle museum, Castiglione d'Orcia, Buonconvento, or Castiglione Fiorentino (the next town over).