The founding of Cortona is seeped in legends which should only be revealed by the locals, but there is no doubt that the town dates back to Etruscan times. Parts of the Etruscan city wall can still be seen today as the basis of the current city wall. If you're interested in Etruscan history, you should visit the Etruscan Academy Museum. One of the most famous artifacts is the Etruscan hanging lamp which dates back to the fourth century BC.
While in Cortona, you should also see the church of the Madonna del Calcinaio. The facade does not appear to by anything special, but the inside has been remarkably restored. It has unusual high arches and they typical Renaissance form of rectilinear forms combined with bowed shapes. Cortona is a great place to just explore as well, just walking along the old paths is fascinating.
You should also take the chance to visit some neighboring towns, like Siena, Orvieto, Montepulciano or Assisi for a fun day trip.