How do I measure speed and manipulate the speed based on distance covered?
They key point here is to know the distance. The simplest way of finding that distance is to know how many times the wheel has completed an entire revolution. Multiply that number of revolutions by the circumference of your wheel and you have the distance the wheel has covered.

Depending on how precisely you want to measure that distance you may get away with a simple magnet and hall effect sensor. Place the magnet on the wheel, and locate the hall effect sensor so the magnet passes it close enough to trigger. Each pulse from the sensor is one rotation of the wheel. 

You could increase the resolution by adding more magnets - 4 magnets set at 90° points around the wheel will give you a pulse for each quarter of a revolution (so multiply pulses by a quarter of your circumference).

If you want better resolution and accuracy you could attach a rotary encoder to the axle of the wheel. This, depending on the encoder, can give you far better resolution than simple magnets. Some will also give you the angle the wheel is at (not that such information would be of interest in this project).