[Prosthetic Conscience]

@Agent Steel , as Saeko said, you build up to the larger distances. Parallax measurements of closer stars, as the Earth orbits the Sun, gives you their distances (and they all fit with the yearly movement of Earth in the orbit - that's how you know it's not the stars); the measurement of some cepheid variable stars with that method gives you the general formula for the absolute brightness of any cepheid variable star, and from the apparent brightness to us, we can get their distances. When those were then observed in other galaxies, they could find the distance to the galaxy (this was what Hubble did, about a century ago). From that, they noticed that the further galaxies were, the bigger the Doppler shift towards the red was for the spectrum of light from them. This shows a relationship between distance and speed away from us, which shows the universe is expanding. So then you can work out the distance of far-away galaxies and quasars from their red shift, which are the objects billions of light years away. And they allow for general relativity in working that out.

[Ter]

The universe is expanding, so why aren't we expanding with it?

If we would expand in concert with the universe, we would not be able to notice any expansion. Since we can, we can confidently conclude that we are not expanding, at least not as much as the universe. QED.

@Agent Steel you can do an interesting experiment with light. Close your eyes and you cannot see the light any more. Is is still there when you cannot see it ? Nobody knows. The simulation we live in will make the light re-appear when you reopen your eyes. That is why Schroedinger's cat is alive and dead at the same time ! Problem solved.

[Agent Steel]

@Prosthetic Conscience

Right, but if the Universe is EXPANDING, then obviously the most distant stars cannot be measured, since they're on the edge of the Universe.

[Prosthetic Conscience]

@Agent Steel , yes, the theory does indeed say there are objects beyond the limit that we can observe. But we can observe galaxies and quasars that are billions of light years away. Have a read of this - it's rather mind-blowing to me, but it may be a start, and there may be books that explain it better: https://en.wikipedia.org/wiki/Observable_universe