Piero della Francesca is one of my favorite painters, along with the Venetian Giovanni Bellini and the Dutchman Jan Vermeer (above, Piero's Saint Julian).
Here is a great little website of Piero's birthplace, Sansepolcro, and the surrounding area. The English version of the site can be accessed by clicking on the little British flag.
The Valtiberina (Tiber Valley), apart from its own significant attractions, is a fine base from which to explore both Tuscany and Umbria, as it is very close to the border of the two regions. Assisi and other Franciscan sites would be a daytrip from the area. It also includes the minuscule village where Michelangelo was born, Caprese Michelangelo.