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This is a question I asked years ago in an astronomy course, but to which I never got a straight answer. Please feel free to correct me if any of my assumptions here stray from facts. It goes like this:

To any observer, at large enough scale the universe appears to be receding in accordance with Hubble's constant, which causes the light that reaches them from objects such as distant galaxies to experience a red shift. Technically, though, if the observer is standing on a planet, they are in orbit moving in a particular direction at any given time, and so the distant galaxies in the direction of that should have a(n extremely) slightly smaller red shift while those in the opposite direction should have one slightly more severe. This means that, relatively, the galaxies behind are receding at a faster rate from the observer than those in front.

If the observer were to possess instruments of sufficient precision to determine the discrepancy in the red shifts and the means to adjust their velocity in any necessary way, would it be possible for them to equalize the red shift in every direction and thus be motionless in respect to the expansion of the universe itself? The cosmic background radiation might be a good reference for measuring red shifts in every direction, since it should be equally distant everywhere and very, very red shifted.

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Yes. This is is actually something that is done in the analysis of redshift data. If no correction to the redshifts are made, then the redshifts are known as "geocentric." When you correct for the motion of the Earth around the Sun, the redshifts are known as "heliocentric" (see, for example, the description of data in the 6dFGS database).

Less common is to correct redshifts for the motion of the Milky Way all the way to the cosmic microwave background's (CMB) frame. This is done by applying a shift to a frame where the CMB dipole is zero. There are two reasons why this is less common. First, the CMB dipole was only well measured relatively recently by the WMAP and Planck missions. Second, correcting for the solar system's motion relative to the CMB fixes only half of the problem. The target galaxy will also be moving with respect to the CMB at speeds that are probably comparable to the Milky Way. This fact leads to distortions in when we try to use redshift as a proxy of distance, some of which are called "Finger-of-God" for their tendency to make plots of galaxies appear to contain filaments that point at the observer.

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You can define an observer to be motionless if they observe no dipole for the CMB. Galaxies are much less useful (than CMB), suffering from shot noise and random motions.

Of course, even the CMB dipole is slightly distorted (from that in a smooth expanding universe), but it's the best you can get.

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