Find a private driver or car rental provider in santorini that is reliable and speaks English. Chose your driver ahead of time so you can ensure that you have a ride available. Plus, you can probably get a pretty good price because you can connect with the driver directly via Synotrip instead of going through a middle man.
Drivers should be able to provide you with a digital copy of their certification, such as a driver's license or tour guide license.
If you have certain requirements, make sure to bring it up with your santorini driver ahead of time. Remember that driving norms might not be the same in santorini as they are in your home city, so it is worth discussing even the more-obvious safety requirements such as seat belt usage and speeding.