The so called expansion of the universe is not as trivial as most people think. What is happening, in fact, is that the distance between two points in space (note that I'm not talking about objects with velocities, but just coordinates in space) increases with time in a manner proportional to a given factor (in this case, the Hubble constant - which is actually not a constant in time, but let's ignore that for now).
This means that galaxies are not exactly moving away from a specific point with a particular velocity (i.e., a particular rest frame), but that the space between them is expanding. It is somewhat hard to wrap your head around the difference between these two situations, but this has to do with the geometry of space, and not with what's inside it.
It is widely accepted that the expansion of the universe is homogeneous and isotropic (the Cosmological Principle), which means that there is no "special" position (rest frame) on the universe, and whatever your velocity and your location, the universe will seem to expand the same way.
The best way to imagine this is to reduce to 2 dimensions expanding in a third dimension. For example, suppose that you live in a galaxy embedded on the surface (a 2D universe) of a balloon (completely unaware of the third dimension), with other galaxies distributed homogeneously. If, for some reason, the balloon is expanding isotropically and, every point of the surface will be moving away from every other point, so will every galaxy from each other. Think about this: is it possible to find a special, "still", point (without resorting to the elusive third dimension) where you will see galaxies moving from you in one side and galaxies moving towards you on the other side?
Our universe works similarly, but we're embedded on a three-dimensional space instead of a two-dimensional one.
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