Galaxies are moving with velocities proportional to the distance from the point they are being observed (or, generally, the frame reference we use). This means, galaxies that do not belong to our universal neighborhood are moving faster than the other in smaller distances.
The vast majority of the galaxies are moving in such a way that increase the distance we measure. This is a strong indicator that our universe expands. The rate of expansion lies in the proportionality of velocity to the distance, which is derived through a very simply law, known as Hubble's Law.
Working with observational data we can easily visualize what is indicated by Hubble's law; the more distant the galaxy, then the faster it is moving away from us.
The images above represent the distribution of galaxies in a spherical region of our universal neighborhood. It is called Local Supercluster, its diameter is a bit more than 100 Mpc and it contains several thousands of galaxies. Every dot corresponds to a galaxy. The center is located to the galaxy M87. The colors indicate different velocities. Red has been attributed to slow moving galaxies and magenta to high speed moving galaxies located in the outskirts of Local Supercluster.
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I created the above visualizations with Mathematica. data are available here (this catalog of galaxies is a portion of the 2MASS catalogue).