The palio is a horse race run in Siena twice each summer, once on July 2 and again on August 16. Attending the Palio is neither easy or economical. Even my friend who has a house just outside of Siena and who is totally devoted to the race never actually goes to the campo to watch.
If you have any interest, you should watch the Palio documentary which is currently available to rent or buy on Amazon Video.
I haven't found anything that is a truly official website on the palio. But this site seems pretty good: https://www.ilpalio.org/palioenglish.htm
The Infiorata takes place in Spello during the celebration of Corpus Domini. The key Sunday is 9 weeks after easter, which in turn is the first sunday after the first full moon after the spring equinox. So like Easter, the date moves a lot. In 2026 the dates are June 6 and 7.
The Infiorata proper happens Sunday morning. The artists have spent the night working on their flower pedal pictures. Saturday and even Friday can be interesting times to visit as people are working on plucking flower pedals and finally making the art work.
If you're visiting, plan to arrive very early (7:00 AM) because the crowds will be coming. And starting at 11:00 AM a procession walks over all of the artwork scattering the pedals to the winds. So the time to visit is short. Also be aware that the streets are steep and a system of one-way streets is put in place Sunday morning.
Official website https://www.infioratespello.it/