How does the doppler effect change wave speed?

It doesn't change wave speed at all. 

It only changes the wavelength perceived by the observer
if he is moving toward or away from the source.

Notice that if one observer is moving toward a source, and another one
is moving away from the same source, then one observer would see an
abnormally long wavelength, and the other one would see an abnormally
short wavelength, both from the same source.

The Doppler Effect ... like beauty ... is in the eye of the observer.