Should we try public transport Siena to San Gimignano or can we hire uber/taxi/ private driver
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