How to use public transportation sienna to San Gimignano

Depends on your budget

I would think a driver will be a couple hundred euro for round trip

No Uber in Italy

You can get an estimate on driver here
www.daytrip.com

Bus is cheap and easy but makes a lot of stops and schedules may not work for you

Faced with the same thing we rented a car. There are buses that will work but the car was quicker and a little more freedom as to when we come and go. Need an international drivers permit if you rent.

The bus line from Siena to San Gimignano is #130. To view the schedule, go to at-bus.it

The #130 bus timetable SIENA-SAN GIMIGNANO is below.
https://www.at-bus.it/en/linee-e-orari/siena-extraurbano-130
You can also download the PDF.
The Siena bus terminal is on Via Tozzi (near Piazzale Gramsci).
It’s about 1h15min journey one way. By car it would not be much shorter (about a bit less than one hour).
If you plan to visit only San Gimignano the bus is the way to go. No reason to spend hundreds of € on a private driver when you can get there with a fistful of dollars. In the way back to Siena you could also visit Monteriggioni (very small walled village). The 130 bus stops there too along the way.

The only problem with the bus is that the schedule either leaves you with too little or too much time at San Gimignano. San Gimignano looks charming and better from a distance from the road or in pictures. Once inside it is small and touristy, I was ready to leave after about 30 minutes