In order to reach and explore the Quantum Moon, the Nomai documented their discovery of the rules of quantum behavior near a series of quantum shards. If you haven't found these locations yet, you can do so by getting close to a quantum shard and pointing your signalscope at it: this will identify the shard in question and save the Quantum Frequency for future scanning purposes. Using the quantum frequency, you can easily see which planets have a quantum presence, and you can even get a sense for the approximate region where quantum shards can be located.
Going by proximity to the sun, it's easy enough to see that Ember Twin, Timber Hearth, Brittle Hollow, and Giant's Deep all give off the quantum signal, as does the Quantum Moon itself. So, which of these is important for our purposes? For each one, I'll provide increasingly specific clues followed by the actual answer—but if you'd rather not use the InvisiClues approach, the "too long; didn't read" answer at the top tells you exactly what you need to do. If you'd rather have more gentle hints, pass on by to the sections below.
The TL;DR answer for those short on time:
Launch a probe at the Quantum Moon and snap a photo of it, or use your probe launcher's photo mode to take a snapshot. Make sure that, whether in your ship or just your suit, a photo of the Quantum Moon in space is visible in your interface; you may then descend to the Moon's surface.
The quantum signal here leads you toward the cliffs surrounding the dry lakebed at the north pole surrounding Chert's camp. There aren't any obvious quantum rocks on the surface, so it would seem that the signal must be coming from underground.
How to proceed, then?
If you poke around in the caves in this area, you may be able to locate the source of the signal, which the signalscope identifies as the "cave shard". This shard can disappear or reappear while you're not looking at it. Odds are good that you may also find Nomai text from Melorae, who is coordinating search-and-rescue for her apprentice Coleus. The text should direct you to check out the cave at the bottom of the dry lakebed at the north pole.
I can't figure out where that is.
As you likely realize by now, the rising sand levels on Ember Twin are the biggest obstacle to any sort of underground exploration, so you'll need to look for this cave at the start of a fresh loop. Land somewhere near the north pole: it's time to get as far down into the bottom of this pit as you can. The passage down here goes farther than you might expect; you'll need to pass through some sandfalls and even ascend a series of rock ledges jutting out from a veritable river of sand. You'll know you're getting close when you find the passage has narrowed to a maze in which the rocks and cacti are change positions while you're not looking at them.
I think I found the destination. Now what?
The cave with the Nomai skeletons has some explanatory text: Coleus was on the cave shard itself when Melorae's lantern was accidentally extinguished. Once the lantern was re-lit, both Coleus and the shard had vanished. To test this behavior yourself, keep your flashlight on but turn the lamps in this room off using the switch here. Try to keep the cave shard in view while you boost yourself up to stand on top of it; once you're in position, turn your flashlight off, then back on. You and the shard are now in a different cave.
I did all that, but I still don't understand the rule itself.
The rule of quantum entanglement means that a person supported by a quantum object can cease to act as an observer if they are in complete darkness. If the quantum object arguably exists in each of its possible states simultaneously, so does the person the object supports, and so by manipulating the light, the quantum object can effectively serve as transportation. Using this rule, you can use the cave shard to travel between at least three caves on Ember Twin, for example.
So, can this quantum rule help me get to the Quantum Moon?
Unfortunately, this quantum rule doesn't really help you land on the Moon, no. The rule requires you to already be in contact with the quantum object before you can use it for transportation, and the problem here is that the Moon's atmosphere cuts your visibility enough that it relocates before you can actually land on it. It will be useful to remember this rule, but not for figuring out how to land on the Moon.
There are actually two separate quantum signals on Timber Hearth: one comes from a crater on the south pole, while the other comes from the crater in which the Hearthians built their village.
Yep. In fact, the village shard is the one you'll likely see on your trip through the observatory on your way to get the launch codes from Hornfels, so you're unlikely to learn anything new from it. If the Nomai had something to teach regarding a quantum shard on Timber Hearth, you'll need to find the shard near the south pole.
So how do I find it?
It's pretty easy to fly in your ship and follow the signal directly to the Quantum Grove, but if you'd prefer to jet your way over in just your suit, it's trivial to locate the south pole using your suit's mini-map. Pretty much every location on Timber Hearth is set up so that you can get around even without your suit, though, so if you'd rather clamber around wearing your casual attire, that's fine, too. Once you're in the Quantum Grove, you should be able to see the shard itself, as it's right on the surface.
...I don't get it.
Well, it's a pretty small area: there are trees, including one tree with a bit of Hearthian poetry on it, there's a geyser, and there's the quantum shard itself. All of these objects can move around the grove at will as your sight-lines change, and you may be interested to note that the lines of the poem rearrange when the tree is in a different position. If you've got your suit, you can also drop into the geyser to explore underwater, but that just takes you to the network of geothermal vents beneath Timber Hearth's surface. If you follow the path out of the grove, you can also find a note from Chert at the top of wooden ramps talking about quantum signals.
So what's the quantum rule here, exactly?
If the absence of Nomai writing hasn't made it obvious yet, this location wasn't among those that they used to teach quantum rules. That's not to say that there's nothing to learn here, though: this is likely to be one of the first examples the player will encounter in which quantum shards cause other objects nearby to exhibit quantum characteristics, which can be beneficial if you're having trouble locating the quantum shards on other planets.
How does this quantum rule help get us to the Quantum Moon?
Pay no attention to my deceptive question! The contents of the Quantum Grove definitely don't help get us to the Moon. I've heard a rumor, though, that you may want to check out the grove shard and then ask Gabbro about it on your next visit to Gabbro's camp. This area is also important for contextualizing something that happens later in the game, so I can't really endorse skipping it.
The quantum signal on Brittle Hollow is easily tracked to a clearing on the Brittle Hollow equator, where the shard is sitting right out in the open. There's plenty of Nomai writing in this area, both at the top of the shard itself and on the structure in the center of the clearing. Be wary of parking your ship in this area; quantum objects can fling it off the planet altogether, to say nothing of damaging it.
...I'm overwhelmed. What am I supposed to be doing?
The Nomai built something called "The Tower of Quantum Knowledge", but like most Nomai structures on Brittle Hollow, it's descending from the surface down toward the core of the planet. Like a traditional tower, though, you're supposed to enter from the bottom and climb your way up. The Tower contains details about a quantum rule which you'll need before you're able to explore the Quantum Moon, but the wall with the gravity lattice has shattered, so you cannot ascend it. Your objective, then, is to figure out how to get to the upper chambers on the inside of the tower.
I can't figure out how to get there!
Let me tell you some things that don't work: you cannot get to the Tower so quickly that the wall hasn't shattered yet. You cannot use your suit's jets to fly to the top of the Tower, regardless of whether you start from the floor or from the wall. You cannot fly your ship into the interior of the Tower. While you can shoot your little scout into the chambers through the gaps in the walls or the ceiling, you can't read Nomai text through your scout's photos. But, putting the scout in the Tower should tell you some information useful for figuring out the correct answer.
Just tell me already.
The scout will reveal that the structural integrity of the Tower is less than 100%: this means that the Tower will eventually fall through Brittle Hollow's black hole. What's more, leaving the scout on the Tower makes it much easier to locate it after it emerges through the white hole on the other side of the star system. This happens close to the end of the 22-minute cycle, so you won't have much time to investigate the Tower's interior, but you should have just enough time to fly your way in and read two scrolls in the uppermost chamber.
What's the actual quantum rule here?
Unlike the other Nomai lessons, the Tower of Quantum Knowledge just tells you its rule outright, which doesn't really have anything to do with the obstacle you had to overcome to get to it. The rule of the sixth location states that there exists on the Quantum Moon a shrine which the Nomai have built, and in order to explore the Moon's sixth location, the shrine must be located on the Moon's north pole.
Does this rule help us get to the Quantum Moon?
While this rule tells us about some of the features on the Moon and a way that we may be able to make use of them, it gives us no information that will help us actually arrive on the surface itself. This was not the rule we needed.
While Giant's Deep is covered in tornadoes, you have probably noticed that a much larger tornado blocks one of the poles. The quantum signal on this planet seems to come from within this massive windstorm.
How do I get there?
That's the puzzle, isn't it? You can't fly through the winds directly. Fortunately, you can either go over under the storm to find an island with a tower on it. This is the Tower of Quantum Trials, and once you drop through the hole in the top of the tower, you'll be presented with a series of challenging quantum trials. After you overcome each challenge, you'll be taken to an intermission floor with a quantum shard on it, and then you'll face another trial. Solve four trials to prove your understanding of this quantum rule.
My goal keeps jumping away from me!
The first thing that you should realize by now is that, as long as a quantum object remains visible to you, it will not move, so the first arch should be trivial to reach; just turn your body so that you continue facing the arch while you walk backwards up the ramp. The second trial complicates this by putting walls around the platforms such that your view of the arch will be interrupted when you go up the stairs to try to reach it, but the Nomai pedestal here should give you a useful clue. See if you can figure this out.
I can't get through these trials.
Equip your scout launcher and take a photo of the arch. (Don't forget the scout launcher has a dedicated photo mode; you don't have to shoot the scout at the arch just to get a photo.) While you have the image of the arch on your screen, you can walk away and lose eye contact with the arch, and it still won't move away. On the third trial, I'd recommend moving close to the crystal such that you can take a photo of just the crystal while you CANNOT see the top of the wall. Then, if you turn away and turn back repeatedly, you can eventually get the arch to appear above the crystal on the same wall. Don't forget to take a photo of both quantum objects together! On the final trial, you need to understand that a particular wall will only ever have a maximum of two crystals on it at once, but you need crystals in three separate locations to make your way up to the top level. Look away and look back until a wall has a crystal on the bottom level and the middle level: take a photo and walk up until you're half-way up the wall. Now, while looking at the middle crystal, get rid of your photo. While keeping the middle crystal in view, shift your view such that the crystal that was on the bottom level relocates to the top level. Take another photo to be safe, and make your way to the arch level. If the arch is on the opposite platform, just turn away and turn back until it relocates to your side. Technically you don't even need to use photography for these latter challenges, if you're skilled.
Wait, so what was the quantum rule?
The rule of quantum imaging, naturally enough! If you don't want a quantum object to disappear when you lose sight of it, you need to take a photograph. It's not enough to just have a camera pointing at the object; don't think you can get away with just planting your scout on it. You need a photo of the object and you need to have the photo of the object visible on your screen. If you're not consciously observing the image, the effect stops working, and the object can move again.
Can this help us get to the Quantum Moon?
Yes, it can. For some reason, the Quantum Moon's atmosphere permits light to reach the surface, yet it blocks your vision enough during your descent that the Moon will relocate. However, even though you can't see the Moon's surface from space, taking a photo there is sufficient to permit the rule of quantum imaging to trap the Moon in its current position while you land.
Somewhere in the preceding text, you should have found what you need to access the surface of the Quantum Moon. If you've accessed all of the locations above, you now have all the information you need to unlock all of the Quantum Moon's secrets. Don't forget to point your signalscope at the planet's surface to identify the signal, and let me know if you have any questions!