I was in Siena August 2014 for the Palio and I'm planning to return in 2017. We spent 4 days there for the race a loved it. Go to Siena during your trip.
You can access any sights in the town on those 2 days before the race just like any normal tourist day.
Twice a day they do practice trails that you can go watch for free in the Campo. They do a dress rehearsal the day before which I would say go to that trail when you are in town. I forgot the times of the trails, you can always ask your hotel staff or check the palio website as it gets closer. You may encounter small parades of the contrada (neighborhoods) walking around the streets singing which is nice to see. Also each contrada does a big outdoor dinner, (like a giant block party) the night before the race. You can get tickets for those dinners. I did one and it was amazing. Felt like we were part of the group when we left, and we had a horse to cheer on for the palio. CAn't wait to go back for 2017 Race !!!
After I was there in 2014, I discovered a few years before Kenny Mayne did a piece on the Palio for ESPN. It's actually pretty good and you can get a feel here for the days before the race. https://www.youtube.com/watch?v=6GYvns0ejV4