According to relativity, mass can never move through the Universe at light speed. Mass will increase to infinity, and the amount of energy required to move it any faster will also be infinite. But for light itself, which is already moving at light speed… You guessed it, the photons reach zero distance and zero time.
Note the words "according to relativity". Relativity doesn't help anyone get rid of those pesky infinities that keep turning up in quantum calculations. Relativity says that wherever or whatever you are, you're relative to something. And in the quantum world, the most significant relationship relating to reality is that of the observer with the subject.
Time has a special relationship with light. For brief elucidation on that relationship, see this article on space-time. And for more fun with Infinity, see here. If you type "infinity in quantum equations" into your search engine, you'll find lots of scientists claiming lots of different things, including the option that infinity doesn't exist.
Hubble wanted a constant, and got one; Hubble Constant = rate of universal expansion. Only, what if the expansion rate is not constant at all? What if it's greater than can possibly be imagined, because we'd have a hard time keeping up? What if we see 'visible light' because we exist at the same resonant frequency as visible light? What if that resonance is the fabric of our very reality, the material on which the laws of physics rests, the only thing we know to relate to? Electrons can, in certain circumstances, travel faster than light, and when they do, they 'live' longer than their counterparts. Speed and time have a strange mode of interloping.
Light speed = a Constant defended with affidavits if necessary. To buck the trend, let's suppose for a moment that we're all moving at the speed of light, along with everything else we can see, and that relativity - general or otherwise - needs to take account of this in order to be able to do something practical with the infinities that keep popping up all over the maths.
You don't need to go anywhere, or even to move from one place to another to exist at light-speed, for if all points in time are simultaneously Here, Now, then relativity can swing all the bats it likes - you and I are still existing in the Constant, which would be how we get to navigate our way around the world we live on. Take the Block Universe theory, and try dallying with Eternalism for a while just to see where it leads. Could be the start of a new kind of journey....