The galaxies outside of our own are moving away from us, and the ones that are farthest away are moving the fastest. This means that no matter what galaxy you happen to be in, all the other galaxies are moving away from you.
However, the galaxies are not moving through space, they are moving in space, because space is also moving. In other words, the universe has no center; everything is moving away from everything else. If you imagine a grid of space with a galaxy every million light years or so, after enough time passes this grid will stretch out so that the galaxies are spread to every two million light years, and so on, possibly into infinity.
