3 days in Siena

We spent 2 nights in Siena back in the late 80s when we lived in Germany and drove down to Italy. You can see the city in one full day. Not sure you need three nights. Have you been to Florence? San Gimiagnano is a small town not too far from Siena.
http://www.sangimignano.com/en/

Also, Pisa and Lucca.

How exciting! I'm glad you're spending so much time there. There is an extensive network of buses that travel all over Tuscany - the big bus terminal in town is at the Piazza Antonio Gramsci. There's a big board announcing all the departures. There is a ticket and information office, located underground - you enter through a small, inconspicuous kiosk at ground level.

I agree with the advice to explore the list of bus routes. There are lots of lovely towns in the region. If you spend one full day exploring Siena, you can use the other two days for bus trips.

Here is a map of the destinations throughout the region served by the Tiemme bus company. As you can see, there's a bus to San Gimignano and in another direction a bus to Montepulciano (just as examples of towns you've no doubt heard of). I understand there's also another bus company called Lazzi.

https://www.tiemmespa.it/wp-content/uploads/2018/12/Sienaextraurbano.jpg

"You can see the city in one full day. Not sure you need three nights."
Strange how impressions differ. In the last 15 years or so I've spent close to four months there, and still discover new things.

"Lucca and Pisa" - are both difficult to visit from Siena. Train times to Pisa are between 90 minutes and 2 hours. Lucca is another 30 minutes + connection times. If interested in either city (and I like both), then it is far easier to visit from Florence.

San Gimignano is lovely, but can be wall to wall tourists. Monteriggioni is lovely, although very small - there is an hourly bus.

Rent a car. Why limit your choices of towns and loose previous time waiting for buses, trains?