In 1929 the astronomer Edwin Hubble measured the velocities of a large selection of galaxies. He expected that about equal numbers would be moving toward and away from us. After all, the Earth isn't a particularly special place in the universe.
Instead, he discovered that almost all galaxies are moving away from us!
Also,Since the time of Hubble we have observed millions of galaxies with better equipment and verified his results. With the exception of a small handful of galaxies close to us, every galaxy is moving away from us.