Yes, you will feel the energy in Siena during the days of the Palio. We were there last July and it is a spectacular event. The only way you will see the race is either to buy tickets (if you can find them) or by being in the center of the Campo (think Time Square). The ticket prices are set by the owners of the bleachers. We were able to get our tickets for 300 euros each, but some were asking up to 600 euros. If you would like to see the horses, but not attend the race, the best time to see them all would be on June 29 at 12:00 pm or Aug. 13 at 1:00 pm when the drawing takes place in the Campo to assign the horses to a Contrada. Even if you do not see the race, the week leading up to the Palio is a very exciting time in Siena.
This website provides FAQ about the Palio. (You may translate it into English by right clicking on your mouse)
https://www.ilpalio.org/domande.htm
This website is by a travel writer who provides a good description of the Palio and Contrade.
https://www.thegeographicalcure.com/post/guide-to-the-palio-of-siena
This website provides the times of the events leading up to and including the Palio.
https://www.ilpalio.org/orari.htm