If we were able to travel at the speed of light, it would certainly enable us to explore vast distances of the universe. However, even at the speed of light, the universe is so incredibly vast that we wouldn't be able to see the entire universe all at once.
The observable universe is the portion of the universe that we can potentially see or detect from our vantage point on Earth. It is limited by the finite speed of light and the age of the universe. The current estimate for the size of the observable universe is about 93 billion light-years in diameter. This means that the light from objects farther away than this distance has not had enough time to reach us since the Big Bang.
As for why we don't see other galaxies moving away from us while they are supposedly traveling at their own light speeds, this is due to the expansion of the universe. The expansion of space itself causes galaxies to move away from each other. This effect is observed as the cosmological redshift, where the light from distant galaxies is shifted towards longer wavelengths, making them appear redder.
The expansion of the universe is not limited by the speed of light, so the recessional velocities of galaxies can exceed the speed of light as measured from our vantage point. However, it's important to note that this does not violate the principle of special relativity because it is the space between galaxies that is expanding and carrying the galaxies apart, rather than the galaxies themselves moving through space at speeds greater than light.
In summary, if we could travel at the speed of light, we could explore vast regions of the universe, but we wouldn't be able to see the entire universe at once due to its immense size. The apparent lack of galaxies moving away from us at their own light speeds is due to the expansion of space itself, causing the observed redshift.