How fast was the expansion during the big bang?

[Bandersnatch]

the beginning the light was already ´here´,

It wasn't.
The oldest light we see (the cosmic microwave background radiation, or CMBR) was emitted about 380 000 years after the big bang singularity (understood here as the limiting point of the theory rather than a physical entity). Before that time, the universe was so dense and so hot that all the matter was in the plasma state, which is opaque to light. The gas that emitted CMBR was 40 million ly away from the gas that ended up as our galaxy.
The oldest galaxies we see needed time to form, and during the few hundred million years the expansion kept on doing its thing, and much more rapidly than it does today. The farthest, oldest galaxy we can observe was at about 3 billion ly distance when it emitted the light we get to see now.

One thing I can think of is that the expansion in the beginning was much faster than the speed of light.

While you're right that the rate of expansion in the early universe was much higher than today's, it makes no sense to talk about the whole expansion being of any speed. It always depends on which galaxy (or more generally, which object) you look at. At any given time in the history of the universe you'll have objects that are receding from the observer at speeds in excess of c.
For example, galaxies currently observed as receding at c lie at the so-called 'Hubble radius' distance, which in terms of proper distance (meaning how far away something is NOW) is equal to ~14.4 billion light years. Those same galaxies, if observed at the time they were forming, would recede at over 3c, and the gas that ended up as those galaxies would be seen receding at over 20 c at the time of recombination (when the universe stopped being opaque).