The expansion of the universe does not occur at the speed of light. The expansion rate of the universe is governed by the Hubble constant, which determines how fast galaxies are moving away from each other on average. Currently, the accepted value of the Hubble constant suggests that the universe is expanding at a rate of about 70 kilometers per second per megaparsec. This means that for every megaparsec (roughly 3.26 million light-years) of distance between two objects, their relative velocity due to the expansion of the universe would be approximately 70 kilometers per second.
While the expansion of the universe is happening on a large scale, it does not affect local motion or objects within gravitational systems. In other words, the expansion of the universe does not directly limit our ability to travel to distant galaxies within our local group or cluster, as the gravitational forces within these systems can counteract the expansion.
Regarding the perception of time, it is influenced by various factors such as relative motion and gravitational fields. For an observer traveling at high speeds close to the speed of light relative to another distant galaxy, time dilation effects would indeed occur. Time would appear to pass differently for the traveling observer compared to an observer in the distant galaxy.
However, it's important to note that the expansion of the universe itself does not cause time dilation effects or prevent us from perceiving time. Time dilation is primarily a result of relative motion or strong gravitational fields. The expansion of the universe affects the overall scale and structure of spacetime, but it does not directly influence the perception of time for local observers within galaxies or clusters.