How to find whether a vehicle turned left or right or didnt turn using gyroscope data
VERY difficult!  You just need data from ONE of the Gyroscope channels, as a vehicle (we hope) only turns around one axis (earth-vertical).  A gyroscope reports angular velocity, and thus to get angular direction, you would need to integrate, and integrating accumulates errors over time, so often that data is high-pass filtered and you lose the mean.

Similarly, from accelerometers you can integrate to yield velocity, and double integrate to yield path, with the same accumulation of errors problem.

You need a magnetomer system.  This will give you direction of the car with respect to magnetic north.  Sometimes, systems with magnetometers are called "9 dof" systems ( https://www.sparkfun.com/products/10736 ) , as they have three axes of gyros, accelerometers, and magnetometers.  You could get an isolated magnetometer ( https://www.sparkfun.com/products/10530 ) , and that and the obd should be all you need.