The Piazza del Campo, the unique shell shaped piazza at the centre of the city, and twice a year the racetrack for the Palio q.v..
The Palazzo Pubblico, Siena's City Hall for almost 800 years, contains amongst many other things the famous frescos on good and bad government by Ambrogio Lorenzetti, frescoes by Simone Martini and Duccio, and access to the Torre del Mangia, from whose top you can view a beautiful panorama of the Sienese countryside.
The Duomo, Siena's magnificent black and white Italian Romanesque cathedral including the Libreria Piccolomini, Baptistery separate entrance and charge, and an attached Museo dell'Opera del Duomo separate entrance and charge (http://www.operaduomo.sie...) which includes the famous Maestà by Duccio. After you've seen all the art, you can also treat yourself to a beautiful panorama from Il Facciatone in the museum.
The Pinacoteca, full of Sienese painting from the city's Medieval heyday.
The Palazzo Salimbeni, built in 1472, is the world headquarters of Monte dei Paschi di Siena, the major banking corporation which has a hand in much of the economic and cultural life of the city. The building itself is well worth looking at from the outside. Inside, there are interesting documents showing the history of banking, plus a collection of paintings and other artwork, but you would have to request permission in advance if you'd like to have a chance to view them.
The house where St. Catherine of Siena was born is just a few blocks from the duomo. This 14th century mystic is one of the principal patron saints of Italy.
The Piccolomini Palace built in 1459 by the well-known architect Bernardo Rossellino, disciple of Leon Battista Alberti. The Piccolomini Palace is one of the most popular sites on the Piazza del Campo. The Florentine style palace is home to the official archives of Siena.