Cortona is a great jumping off point for Tuscany and Umbria. We have been there five times and are going back next year.
Montalcino, Pienza, Montepulciano, Monticchiello - These towns are close together and make a wonderful day trip. There is a gravel road, "strada bianca" between Pienza and Monticchiello that is often featured on calendars of Tuscany.
Siena - wonderful hill town. easy hour to hour and a half drive from Cortona.
Florence - easiest way is to go by train. make reservations for the Uffizi and Accademia.
Orvieto and Civita - another great day trip.
Assisi - hire Anne Robichaud as a guide, we have done the Assisi and wine tasting in Spello TWICE. it's that good.
Lake Trasimeno (sp?) and Castiglione del Lago.
I have been to San Gimignano, and I'm glad I saw it but it's not my favorite.
Cortona is a great town for wandering. Lots of good restaurants too. Outside of town is Le Celle, a monastery where St. Francis stayed, it is a great place to visit