Though on one hand speed of light is said to be the universal limit to the speed, but on the other hand it is also said that just after a few seconds of big bang, the universe had spread over many light years. Don't these two concepts contradict?
You can't move through space at faster than light speed, but expanding space doesn't actually involve anything moving.
The space between things can be growing at faster than light speed, but the things themselves don't actually move through space at all in that process, so their speed is actually 0.
It just looks like they're moving, because the space in between is expanding.
Yep, that's basically it. Moving through spacetime is restricted to c (speed of light) but spacetime itself can expand at any rate. Indeed the red-shift data shows many galaxies 'receding' from us at v>c.