How fast does rocket fuel leave the rocket?

This depends on the fuels used (Liquid Oxygen, Hydrazine, etc) the design contours of the nozzles (if we are talking about liquid fuel rocket engines, and pressure pumps, turbo-chargers etc in the rocket body. The velocity of the rocket is dependent on a good many factors including the chemical reactions of the fuels themselves, the specific impulse as it is called, and as with autos, transmission losses due to plumbing and pipes.