The laws of physics are constant for any non accelerating observer. The speed of light is constant for all observers regardless of the motion of the light or the observer.
Most are familiar with the doppler effect when a train horn passes by and the frequency changes as it passes. A similar effect is seen in astronomy with red shifted observations.
After compiling catalogs realized the red shift was proportional to the relative distance to the object. It was also realized the red shifted objects were moving away. Over time as spectrum analysis became more sophisticated it became possible to determine the relative movements of nearby stars and galaxies.
The absolute distance from one place to another does not consider the movement of stars. The relative distance is more complicated. The distance to the outermost galaxies can be larger than the relative size of the universe due to the effects of relativity.
The constancy of the speed of light was motivated by Maxwell’s theory of electromagnetism. The derivation of special relativity depends on several tacit assumptions (made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.
Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0. To find the relation between the times between these ticks as measured in both systems Δt’ = λΔt satisfying Δx = 0.
GPS satellites with an atomic clock on board were discovered to exhibit slight relativistic effects. This required receivers to consider relativity when listening to the sentences to determine latitude and longitude accurately.
Consider a rod that is moving away at an accelerating pace. It will appear to be shorter as it approaches the speed of light. From the rod’s point of view it is still the same length as it ever was.
Time dilation is explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system. These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the first. So the rod length appears to shorten and the passage of time appears to shorten.
Given a spacecraft that is capable of maintaining 9.8 m/s² then after about 3-4 months time the stars will begin to appear to come from the nose of the spacecraft and time will have slowed so much that it is possible to reach another galaxy such as Andromeda. The voyage from the Earth’s point of view will have taken 700 million years so there is nothing to come back for. Such a trip is one way into a far distant future.
Time dilation exists when an object is very close to the speed of light. The extreme dilation can allow any distant galaxy to be reached but the most distant ones may not look like they did when the voyage began due to the enormouse passage of time. Very distant galaxies may be a large black hole by the time a traveller arrived.
From an outside observer the spacecraft will appear to be using an infinite amount of energy. In fact it a spacecraft cannot exactly reach the speed of light as the dilation effects become increasingly apparent.
Sending a large population on a voyage to distant locations is viable. Discovery of habitable locations is not a difficult task once star charts are assembled.