If you're travelling with Ryanair, buy a ticket for the Autostradale bus service on board the plane. Otherwise buy your ticket at the Autostradale Information Desk. (€15 single. Don't buy a return ticket as it's only valid for one day. Coming back you can buy your ticket at the railway station).
Go to the Autostradale bus stop just outside the arrivals area. Take the 1615 bus to Firenze (Florence) railway station, arriving at 1725. Transfer to the 1730 fast service to Siena, arriving at 1845.
Coming back, take either the 1110 or 1150 fast service from Siena to Firenze (arriving 1225 or 1305 respectively), transferring to the 1315 airport service, arriving 1425.
If those times aren't right for you there are alternative service listed here:
http://www.autostradale.it/pdf/festivitaaereoporti/Pisa-Siena24.pdf
Alternatively take a bus to Pisa Central station (every 10 minutes, journey time 8 minutes) and then travel by train to Siena. The journey takes about an hour and three quarters, including a change at Empoli, with two services per hour. Train times and fares can be found here:
http://www.trenitalia.com/
(Change the the language to English at the top right)