Since the Earth is revolving around the Sun, the Sun is revolving around the center of the Galaxy, and the Galaxy is moving throughout the universe, no doubt the Earth has moved a very long distance, perhaps thousands of light-years, since its formation. Do we have any way of knowing how far?
If you consider the orbiting of the earth around the sun to be a rounding error, the earth has traveled appx 2.9 million light years.
This is using 18 cosmic years of the sun going around the galaxy at at distance of 26,000 light years from the center.
And assuming I didn’t do a math error.
It depends on your frame of reference. Just_Asking_Questions seems too put the (0, 0, 0) coordinate at the center of the Milky Way, and assuming his numbers (26,000 ly and 18 cosmic years of the sun) are correct, his calculations seems legit for the sun.
If you just calculate the distance travelled around the sun where the sun is the coordinate origin I get ~350.000 light years. I am tempted to simply add both numbers together: 3.25 million light years.
But if you start your coordinates in a strongly red shifted galaxy far, far away you could get a much bigger number.
Yes, I think otherwise the question gets too complicated.
The earth moves around the sun, the sun moves around the galaxy, the galaxy moves, the universe expands…
Do we mean “moving” as in total miles traversed, rotating around the sun, and rotaint around the galaxy, and moving through space?
The total distance the earth moved around the sun, again assuming I can do basic math correctly, is 447000 light years. Seems like a rounding error. 
If you want “moving” to be how far across the universe it has moved since it formed, from a point >>over there to where we are now, well, the galaxy is moving at a (put you seat belts on) blistering 1.3 million miles per hour, relative. Over 4.5 billion years we’ve moved about 8.7 million light years away from our starting point.
Should we add all those together? My assumption is the total “linear” distance traveled has less meaning as the universe is expanding during that time. And finding a stable reference point is, well, problematic. The 1.3M mph is relative to the background radiation.
I’m actually surprised at how large the number are! 4.5 billion years is a long time!
Just adding the distances isn’t going to get the right answer. Most of these coordinates from random Internet searches, so caveat emptor.
The problem is that we are not living in a flat universe. The plane of the solar system is tilted about 60 degrees to the galactic plane. Worse, the galactic angle the plane makes to the orbit of the sun around the galaxy isn’t constant since conservation of angular momentum means the Sun’s orbital plane always faces the same direction, whist the Sun’s orbits the galaxy. Overall this reduces the contribution to distance traveled by the Earth’s orbiting quite a bit. And it gets worse.
Movement relative to the CMB is 390m/s, but 620km/s towards the Great Attractor - towards somewhere 30 degrees above the galactic plane and at 272 degrees galactic longitude. Which is very close to edge onto the Earth’s orbital plane. All of which makes aggregate distances messy yet again. If the orbital velocity of the Sun is about 200km/s the Sun describes a flat spiral in space along the Great Attractor direction and the Earth’s orbit contributes a very nearly flat spiral motion (ie nearly zigzag) onto that.
The various flattening of spirals reduce the total distance travelled. The contribution from the Earth’s orbit is probably nearly halved. The contribution from the Sun’s orbit probably drops to three quarters. Very back of the envelope spit balling.
All of this assumes that the next higher component of motion is large enough that there is no retrograde motion. Which seems safe. Retrograde loops would really mess with things.
Alignment to the CMB reference frame is going to dominate any calculation. I have no idea if there is work trying to work out a history of the local cluster’s motion. It may be too chaotic to usefully try. Would be interesting if anyone has tried.
That figure is a long way from accurate. You cannot just use dπ to calculate the sun’s galactic orbital traversal, because it bounces up and down in the galactic plane due to gravity effects from the plane itself. I am not sure how far or steep the bouncing is, but I suspect it is more than a rounding error.
The whole thing is an approximation! We say the earth is 4.5 billion years old. What if it is closer to 5? We think we’re 26000 light years from the center of the galaxy. What if it is 27000? The galaxy is moving at 1.3M mph. Are we sure?
And of course the caveat that Special Relativity denies that there’s any fixed “grid” of space, an absolute framework that could be used to declare either a position or a velocity.
SR or GR doesn’t stop us from using the CMB as a reference frame. It just stops us making weird noises about it being a privileged one. The CMB may well be a sensible place to start thinking about how our patch of the universe evolved.
I don’t think there is a right answer. Recall Einstein, where all is relative: in terms of two particles/celestial bodies, all you can say is that they are A distance apart at time Y, and then B distance apart at time Z. Unlike on a 3D surface like the Earth, there is NO absolute “position” in space (“Eureka! This is the center of the universe! And they called me mad, hahahaha!”) on which to base your universal coordinate system on that you can use to backtrack to this hypothetical point (Coke to Lumpy).* IOW I don’t think it is even possible, in principle or reality, to hop in your mega-warp-drive capable starship after doing all of your Spockian calculations, and travel to THE exact point where the Earth first came into existence (ignoring the additional issue of defining the “instant” when Earth first “officially” existed). Especially as space has been stretched during all of that time too, as also indicated above.
As David Gerrold once said, it would be like trying to bisect a sneeze. Not even wrong, IOW.
But I’ll summon and defer to the master since he hasn’t deigned to materialize in this thread yet. @Stranger_On_A_Train
[*Even the surface of the Earth is in constant flux at small scales, and over a long period of time our current latitude/longitude scales will become increasingly inaccurate.]
I’m hardly “the master” when it comes to cosmology, but I think @Francis_Vaughan has already given the closest thing you will get to a definitive answer on the topic (although I think there is a discrepancy in the rates relative to the CMB and Great Attractor, noted below).
As noted above, any linear metric of speed also has to account for those components of rotational motion; as an example, if you were to measure the speed of a point on the surface of the Earth relative to the Sun (which you have to do in ground-based solar astronomy) then you have to add the speed of rotation of the planet (at what ever point it is at in the day) to the orbital motion of the Earth about the Sun, resulting in a epicyclic term (one which has a sine component relative to position or time) in addition to the equations of orbital motion. This relatively straightforward case can be solved analytically but becomes very complicated, and furthermore, because the Earth is actually a couplet with its oversized moon, you also have to add that contribution down to some level of precision. The same is true for perturbations of the Solar System’s ‘orbit’ around the galactic center, although this does not precisely obey Kepler’s equations, and then the motion of the galaxy within the Local Group of galaxies, et cetera. Since we only have a rough idea about the motion of our system through the galaxy throughout its ~5 billion years (even though we know the age of the aggregation now known as Earth as 4.54 bya to within ~50 myr) all of the small perturbations get mushed out in the larger uncertainties anyway.
At the grossest scale, motion relative to the Cosmic Microwave Background (CMB) will dominate because all other speeds are due to rotation and are thus limited by the the tension between ‘local’ gravity holding systems together and the centrifugal component pulling them outward, whereas there is no limit of linear speed relative to any reference frame short of the speed of light. The most ‘universal’ frame of reference is the CMB, which is the stochastic motion of photons first emitted when the universe expanded enough to become transparent (which is presumed to have occurred at the same ‘time’ everywhere, approximately 380 kyr after the singularity event). Since these photons were emitted at very close to the same frequency, they can be used (with some corrections) as a statistically ‘fixed’ quasi-static background, essentially a ‘noise floor’ to any electromagnetic measurements of the cosmos. However, measuring speed relative to them is a tricky process since it has to be done using frequency analysis against relativistic motion. The peculiar velocity of the Miky Way galaxy is 369.82 ± 0.11 km/s relative to the comoving background of the Hubble Flow per the 2018 estimates of the Planck collaboration (earlier estimates were ~600 km/s), largely due to apparent motion toward the aforementioned (and unseen) Great Attractor, which itself is moving at some high rate of speed toward an attactor in the Shapely Supercluster. So, the ‘simple’ answer is that we are currently moving ‘about’ 370 km/s, and have probably been moving at some substantial amount of that speed for most of the Earth’s existence, and thus about 5⋅1019 km or about 5 million lightyears relative to the comoving background. Any attempt at greater precision is going to have uncertainties that swamp more significant figures.
Also, because nobody has curiously yet cited it, Monty Python provided the definitive answer over thirty years ago even if their units are not in standard astronomical conventions and some of the figures are somewhat out of date:
Stranger