The easiest thing I can think off is to monitor the speed of the motors. Ideally you want to use DC motors with a gear reduction. Put a disc with holes on the motor shaft (before the gears) and use an IR sensor to monitor the revolutions (IR sensors should monitor the holes in the discs)
If you have a few thousand RPM you can use an easy PID system which ensures the motors will have an exact speed as well as as a proper acceleration curve
A 16 bit gyroscope is also a very good idea for this. Even with a nice acceleration and deccelaration curve, your wheels still can slip (differences between the motors like friction) which might let your car drive a degree off course. With use of a 16 bit gyroscope you can detect these 'distortions' and allows you to compensate for it.
This is the easiest (with exception of sinmply digging wires under the floor) you can accomplish.
Alternatively you can use stepper motors, but electronically it is not a very efficient solution (power consumption). Though you get to controll your motors with great accuracy.
Having a gyroscope also makes the IR sensors and motor tracking a bit redundant. It is possible to steer your DC motored car purely on the Gyroscope.
Problem with the gyroscope is getting your initial bearings. You could point the car in the desired directions, press a button and it will travel in that directions. But Ideally you want some kind of reference. And I am not having an idea at the moment. I think I would use one of them fancy pixy cameras and put 2 recognizable objects on the wall so your car can get it's bearings.