If we could travel at the speed of light, we would not be able to see the entire universe in the way you might imagine. This is because as an object approaches the speed of light, time dilation occurs, and from the perspective of that object, time appears to slow down. As a result, distances in the direction of travel would also appear to contract. This phenomenon is described by Einstein's theory of relativity.
Since the universe is vast, and light takes time to travel across large distances, even at the speed of light, it would take billions of years for light to reach us from the farthest observable regions. As a result, if we were able to travel at the speed of light, we would only be able to see the parts of the universe that were within our observable range at the time we started traveling. The rest of the universe would continue to evolve and change, but we would not be able to perceive those changes until light from those regions reached us.
Regarding the observation of other galaxies moving away from us, it is true that many galaxies in the universe are moving away from us. This observation is a result of the expansion of the universe. However, it's important to note that the expansion of the universe is not due to galaxies moving through space at speeds close to the speed of light. Instead, it is the space itself that is expanding. This means that the galaxies are not actually moving through space; rather, the space between them is stretching.
The expansion of the universe causes the light emitted by distant galaxies to be stretched as well, leading to a phenomenon called cosmological redshift. This redshift causes the light from those galaxies to be shifted towards longer wavelengths, which makes them appear redder. By studying the redshift of light from distant galaxies, scientists have been able to determine that the universe is expanding.
In summary, while traveling at the speed of light would allow us to see a portion of the universe within our observable range at the time we started traveling, it would not provide us with a comprehensive view of the entire universe. Additionally, the observation of galaxies moving away from us is not due to their own motion at speeds close to the speed of light but rather the expansion of the universe itself.