How would I get from Pisa airport to Siena by public transport.
You can take the train. It takes just under 2 hours.
Try with google mpas: https://www.google.com/maps/dir/Siena,+53100+Province+of+Siena/Pisa,+Province+of+Pisafirstname.lastname@example.org,10.584362,10z/data=!3m1!4b1!4m13!4m12!1m5!1m1!1s0x132a2cbf34bf5313:0x5d731212f12343e3!2m2!1d11.3307574!2d43.318809!1m5!1m1!1s0x12d5919af0f6598f:0xaab80fb5a78478c8!2m2!1d10.4016888!2d43.7228386?hl=en
and select train, or you can visit here for bus http://www.sena.it/Home/78-1-en.html