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.
A solid fuel rocket engine is just what it sounds like. The fuel inside the rocket is completely solid. In a liquid fuel rocket the fuel is a liquid. Bottle Rockets that you can buy in many stores are fueled with Solid Fuel. Most liquid fuel rockets contain 2 different types of fuel. Once that fuel is mixed and ignited you get your thrust.