First of all you have to keep in mind that the speed of light c is a constant everywhere and under any circumstance. This statement comes from Einstein's theory of special relativity, where he postulated that the speed of light is the same for all observers, no matter what their relative speeds. Einstein was able to verify the accuracy of his theories (it was on the solar eclipse of November 1919) through an experiment where he calculated the rays of the light coming from the eclipse to be bent by gravity. But the speed of light c was still a constant. It is true that nothing ever overtakes a light beam.

It is also true from Einstein's theories that when an object approaches the speed of light, it experiences length contraction, and time dilation, so there are the space s and time t that change their magnitude, but the light speed keeps constant.

When Einstein referred to the speed of light, he meant the light everywhere. The space is expanding, the galaxies are moving. The galaxies do not expand; it is the distance among them that increases. Think about a muffin with raisins inside, when you bake it, it expands, but the raisins remain the same size. The muffin is the universe and the raisins are the galaxies. There is also the analogy of the inflating balloon for understanding the expansion of the universe. The galaxies on the surface of the balloon are effectively at rest, and yet as the universe expands, the distance between any two galaxies increases. The galaxies themselves do not increase in size. The rate at which the distance between galaxies increases follows a different pattern discovered by American astronomer Edwin Hubble in 1929. From his discovery, there is the Hubble's law which states that the recession velocity of a galaxy away from us v is directly proportional to its distance from us d, being H the proportionality constant, known as the Hubble constant:

v = H d

Hubble's law describes only the average behavior of galaxies. As a consequence, it predicts that galaxies beyond a certain distance (known as the Hubble distance), recede faster than the speed of light. This result has confused generations of students. The solution is that special relativity applies only to normal velocities -motion through space. It is a general relativistic effect and is not bound by the special relativistic limit.

Having a recession velocity greater than the speed of light does not violate special relativity. I say it again; it is still true that nothing ever overtakes a light beam.

I recommend you a nice article that explains much of your questions in Scientific American of March 2005. Misconceptions about the Big Bang.