Duomo, including Crypt, Museum, Bell Tower and Museum.
Piazza del Campo; City Musuem
Museo di Palio- tells you about the Palio, horse races and the meaning of it all.
Great food on the Piazza
Have a look at these two websites and you can get an idea what is there and what to do/book: https://www.visittuscany.com/en/ideas/10-things-not-to-be-missed-in-siena-00001/
A day trip (or half day) for wine tasting: Montalcino is a famous and wonderful wine, and the vineyards are very nearby. If you like a wine tour, this would be a great excursion from Siena. (also available from Florence.) https://www.viator.com/Siena-tours/Food-Wine-and-Nightlife/d944-g6
Also for Siena and Florence, day trips to Pisa, San Gimignano, and Lucca are very do-able and I would encourage you to see them if you can.