I'd go to Siena for part of the Palio activities. I was there in August of 2014 and plan on going back next year for the July race.
Yes there is more of a crowd the days before with the largest being on Palio day. The Duomo is open, though after the Palio the winning neighbor marches to the Duomo and inside to sing and give their thanks to the Madonna. The Civic Tower on Il Campo may have issues getting into when there are practice runs and not open during the race itself.
The Day before the Palio on the 15th there is a practice trial run at 9am, and then a Full Dress Rehearsal around 6-7pm sometime (forgot the exact time). If you won't be there for the Palio, i'd do this day tour the sights in the morning and watch the dress rehersal (although the horses more so just jog around but people are dressed up). At night each of the Contrada (Neighborhoods) have a big dinner for the whole Contrada which you can attend some of them. If you do it was what seemed like an endless supply of wine for hours at the one we attended.
The 14th also has practice around in the main square.
On Palio day (16th) there is a final practice in the morning. Then early afternoon (think 2pm-3pm) each Contrada brings the horse to their district church for a blessing, followed by flag throwing and a parade to the Duomo. All the Contrada's march around the Duomo and flag throwing for the bishop. They are all dressed up in medieval attire. We saw a little bit of the procession at Duomo cause we went inside the main square for the Palio and they close the gate to cross the track at a certain time. So we had to be inside the center by then. The procession leaves the Duomo and heads to the Il Campo so you'll see more flag throwing around the track. The Palio was around 7pm but we had several false starts.
The center of the campo is packed with people but it was fun to be in there like a can of sardines. The dress rehearsal the day before was about as tight but other practices wasn't that full.
All during the day there will be different contrada marching and singing in the streets. It's a great time to be in Siena for.
Kenny Mayne actually did a nice piece on ESPN several years ago about the Palio. You can watch it here: https://www.youtube.com/watch?v=6GYvns0ejV4
