Siena vs. Florence - the eternal battle which began as an intense rivalry dating back to the 12th century. Throughout it's history, Siena has been at odds with Florence - sometimes triumphant and sometimes defeated. Today, Florence is the heralded city of Tuscany and for many travelers Siena is a quick day trip from the greater city of Florence. From my experience it is rather difficult to truly experience Siena this way.
Don't write me nasty messages defending Florence - I'm not trying to downplay Florence in any way. Florence is a wonderful city and a must-see on any list of Italian cities. If you only have 10 days and it's your first trip, see Rome, Venice & Florence & skip Siena.
However, if you have more time or it's your 2nd, 3rd, 4th trip, spend a few days in Siena. This is the way to truly experience the city. Stay in a hotel with a balcony view. Sip coffee in the morning on your balcony (try for a room on an upper floor for the best view) and watch the scene below you. Even if your balcony just has a view of the street below, watch the goings on.
Poke through the shops throughout the city - it's easy to spend a lot of money here, so be careful. Get gelato from as many diffent places as your stomach can handle. In the afternoon, sit on the Campo for a while and people watch. Visit the Duomo, the Dominican church at the end of the city (I can't remember it's name, sorry!), climb the Torre Del Mangia tower. (Allow some time to climb the Torre, as the spiral staircase is VERY narrow, and you will have to stop and press yourself against the wall as other people go up/down, and they will have to do the same for you. If you're highly claustrophobic, the tower is probably not for you.)
For dinner, head away from the main streets and the Campo and look for a small restaurant filled with locals. Bring your phrasebook, because the waiter may not speak much English. Do your best with the menu, point, smile and enjoy. Take a walk around the Campo in the evening as the sun sets.
Siena is a wonderful city to experience in many ways. If you want to see some sights - it has them - churches, museums, a baptistery, the Torre, of course the Palio (July 2nd and August 16th).