How do we know that space is expanding? How do we know we are moving away from the centre of the universe?

The universe is thought to be isotropic, meaning it looks roughly the same from every point within it. There is no center because the Big Bang created space itself, so everywhere must be considered the "center". The expansion rate acts as an opposing force to gravity, but only at distances of several million parsecs. That is because the force of gravity decreases with distance, while the force of expansion increases with distance. So at some critical distance, expansion will overpower the gravitational pull of galaxy clusters. But within this distance, they will still be gravitationally bound.