When we would get into a very fast spacecraft and speed up, we would still observe the speed of light to be exactly the same. If we move towards the source light will not speed up, if we move away from the source light will not slow down. It will remain exactly as it is.
When we are at fast speed and use a light clock to measure time (by bouncing a photon between two mirrors) we would see exactly the same speed, each bounce would be identical.
Yet when we look from outside our spacecraft, for example from earth, we would see that the clock runs slower.
When we would travel at nearly light speed we would see that time seems to have stopped from the perspective from the outside observer.
And for us the first picture still applies, we observe only straight bounces, each indicating the same time frame.
When we would travel at exactly light speed time would stop for us.
There would be no bounce between the mirrors anymore.
More on this subject on the wiki on: Time dilation
Now suppose we are travelling on a photon which was send out shortly after the Big Bang. It would have travelled over 13 billion years and we would only observe it right now. Before that it just travelled through empty space and from the perspective of the photon it would be the moment it was generated. Time stopped right after it. So while it travelled 13 billion years we can also say it's not even a millisecond old.
The only way we as outside observers would know that it's this old is because it's extremely red-shifted, an indication the source is moving away from us at high speed. Check out Cosmic microwave background
And here we are. Sitting looking at the stars. We see the light of stars that might not even be present anymore.
The light from the Andromeda Galaxy travelled some 2.5 million years to reach us:
And yet when the photon hits our eye it does this the exact moment that it left the star, from the perspective of the photon that is. We are looking at the past and present at the same time. And that's really odd.