Palio in Sienna

If you are there the day before, you are also on time to see the 4th and 5th trial (the previous trials are on the preceding two days). The morning trial start at 9am, the evening trial (last trial) starts at 7:45pm. (the 5th and last trial is called the Prova Generale, or general trial). You should try to see those. After the Prova Generale, each Contrada (district) organizes a street Dinner, called Cena della Prova Generale (General Trial Dinner). I've never had a chance to participate because tickets go quickly, but if you can go. The district assigned horse will eat at the head of the table. These are the addresses of each district society. http://www.ilpalio.org/societA_di_contrada.htm The morning of the Palio, there is another trial, called Provaccia (bad trial). The best spot would be in one of the terraces of the residents or the scaffolding, but you have to pay dearly. Also there is no ticket office where you can buy it. You need to look on the spot or ask your hotel. But I've always seen the Palio from inside the Piazza in the Conchiglia (shell) which is in the middle. That's where the locals go, so it will be very exciting, although it's hard to see since it's very crowded, especially if you are short. Below is the schedule for July 2.
http://siena.guidatoscana.it/palio-di-siena/date-del-palio-di-siena.asp

Thank you Roberto! This is something that I have wanted to see for 20 years.

It's either pricey or free. The seats and balcony spots are quite expensive. The center of the campo (plaza) is free. It slopes up a bit, so you can try to get there early and stake out a high spot. Make sure you get into the center before they close it off for the race, which runs around the perimeter. When we were there it was insanely crowded, smotheringly hot, completely crazy, and totally worth it!! It's a once in a lifetime experience for most of us and we loved it. Take lots of water and sunscreen, and settle in for the spectacle.

Diana, Be sure to "stake out" a good seat well in advance, as the Palio race itself only lasts for a few minutes. Given the fact that it will be enormously crowded, be sure you're wearing a Money Belt and be vigilant for pickpockets. It's going to be crowded, noisy and chaotic so perfect conditions for them to operate. Happy travels!

Roberto has laid out your traditional options really well. The bleachers and balconies are really expensive, and standing in the square is free but very tiring and usually hot. Ive done both. However, there's a third way You go to one of the restaurant/bars ringing the square (like Bar Palio at the top of the square, near the Fonte Gaia) around when the historical parade starts. You can watch everything on their televisions (which cover it perfectly) and HEAR everything that is happening just 50 feet from you. All for the cost of a few beers. I know it sounds like a lesser experience, but it's better than standing in the square, I assure you, and a better bargain that $300-$600 for a seat in the bleachers. Doesn't cost you anything to try. All the restaurants and bars around the city will be showing the palio on their TVs, but the establishments around the square allow you to hear the quiet and the roar and really feel what's happening. It's not a bad way to experience it, really. And when the palio ends you can run out and join the celebration, and then follow the winning contrada to their church!