I understand that we can observe far away galaxies to get an idea how the universe looked like in the past.
Assuming the universe is 13.82b years old, would it be correct to say that if we looked at a galaxy that was 13.82b ly away, we would see whatever happened just after the big bang? If this is correct, what would we have observed had we looked at the same point 1b years ago?
Is this an application for special relativity?
In a static universe it would indeed be true that if you looked at an object, say, 10 billion light years away you would be looking at it as it had been 10 billion years ago. This isn't really an application of special relativity and is merely a consequence of a finite speed of light.
Our universe, however, is expanding and so you can actually see across distances that are somewhat larger than simply elapsed time times speed of light. This is because the space expands while light is travelling from a distant object. Thus, the radius of the observable universe is around 47 billion light years. This is an application of general relativity.
So now we can see radiation that comes from within a sphere of radius 47 billions light years. If we were living a billion years ago we would only be able to see radiation coming from within a sphere with radius of about 43 billion light years so we wouldn't see some of the galaxies that we are able to see now.
In terms of things happenings right after the Big Bang, galaxies of course didn't appear right away. So if you are looking at a galaxy that means that you are looking at something that happened a few hundred million years after the Big Bang at the earliest. The closest time to the Big Bang we can actually observe is about 380 000 years after it because before that the universe was so hot that it was opaque and light could not travel. The Cosmic Microwave Background radiation comes from this time when the universe was a very uniform hot (a few thousand degrees Kelvin) soup of particles.
We cannot see anything closer than 380,000 years after the big bang because that is when radiation and matter decoupled. The CMB is a picture of what the universe looked like at that point. All clumping of matter into stars, galaxies, etc has occurred since then. If we had looked 1 billion years ago, we would see the same except that the CMB temperature would be a little higher.
Philo's answer is spot on, and I'll basically be rephrasing it here into a form that makes more sense to me. Hopefully it will help some others as well.
Rather than only dialing back the clock 1B years, let's go waaaay back and see what things look like: we go back 13.82B years and look out into space... And there's no space! The universe is very (infinitely?) dense and everywhere you look there's basically just energy. Depending on the exact timing, there may or may not even be subatomic particles yet, but it doesn't matter: even in the later stretches of the very beginning, the whole universe is like the interior of a star, so dense that photons leaving one emitter are pretty much immediately absorbed by something else. It's like this for the first 380k years or so. The universe is expanding this whole time though, so the overall temperature is dropping.
Finally the universe expands enough that when a photon is emitted, it has time to go somewhere before it hits something else, and on the flips side of that coin, atomic matter can start to form because its not being constantly plasma-ized by photons. So fairly suddenly, the universe goes from being hot and opaque to being slightly less hot and transparent. All those same photons as before are flying around, but as I said, they now travel pretty freely.
So out the window of your time machine you see a cloud of suddenly free photons rushing off to hit whatever eyeball is nearest, a.k.a. yours; and because the speed of light is finite, you see photons sequentially in time based on how far away from you they were emitted. So the first ones you see are from just outside your window, and then the ones behind those, and the ones behind those... You see a receding "surface" of glowing plasma, because you can only see it as quickly as the photons reach you. You can also see the new matter now, but also only as fast as the light scattering off of it can reach you.
So as time progresses, two things are apparent: 1. You can see farther and farther into the presumably infinite universe - that is, your observable universe is growing larger at the speed of light. 2. Since the light takes time to reach you, the things you see happening "at the edge" are the things that were happening right as the universe became transparent. Meaning there's a direct correlation between how far away you can see and how long ago you can see. Time and distance become interchangeable, in a way, related to each other by the speed of light. Which makes sense, since a "speed" is a ratio of time and distance.
Now there's one little adjustment that must be mentioned, and that is that space itself is still expanding. So things that happened x many millions or billions of years ago look more than x million/billion light years away, even though light still only traveled light speed to get here from there. That doesn't change the fact that you can only see so far away or so long ago, only the ratio of how one relates to the other. Overall, the conclusion is unchanged: as rewind the clock, things look continually closer and younger until eventually they're right in your face and brand new.
