How do you tell which way a wave is moving?

It depends on which particles are moving up and which are moving down. If you have a peak in the wave like this

you can't tell which way the wave is moving. But if the particles are moving up in the left side and down in the right side,

the wave is moving to the left, because the particles on the left side are moving up and hence will be at a peak soon. Conversely, if the particles on the left are moving down and the particles on the right are moving up, the wave is moving right.

Of course, this applies throughout the whole wave, I just looked at a small piece to make it simpler.

This answers "How does the universe "know" which way a wave should move?".

First, consider that "the universe", and "a wave" are mental ideals. They are not natural objects, it is our mind that is used to build objects according to our percepting and reasoning capabilities.

In fact, there are no objects in the universe. An atom is just an ideal. Out there, in the reality that is hidden to our senses (what Immanuel Kant called the thing in itself), there is certainly something that produces what our senses register. But the limits between what is the object (e.g. the wave, or an atom) and what is not the object (the rest of the universe, excluding the wave, or the atom, in the second case) are subjective. Objects exist only due to subjects. An object is not possible without a subject.

But perhaps the essential pattern that we perceive, something that Kant would call a priori (a kind of knowledge that is not obtained from perceptive experience, but created in our mind in order to allow perception), is the object, or simply the thing (assuming that we, rational individuals, are the subject). And, following such a priori knowledge, things contain smaller things and are part of bigger things, as a wave is made of particles in movement. The systems theory is useful to follow this perspective; following it, all systems are made of sub-systems, and are part of supra-systems. We need of such principle in order to understand nature, in order to survive. But for Kant, such approach is not necessarily true. Kant considers that space is also a priori knowledge, and that the fact that things are made of smaller things an error of our perception, leading to rational contradictions, an antinomy for him. (Second antinomy: [1])

Well, waves are not only particles in movement, because in such case thy would behave like particles of dust released out of the atmosphere and far from gravity; waves are particles interacting, like all things in the universe. A rock is such thing because it is not only a bunch of atomic sand; it is a group of atoms interacting (forming molecules, which interact forming an object which tends to keep a shape). Amazing, it is!

And, in case of your example, the universe (which is a group of parts invented by our minds) is not telling the atomic particles how to behave, like no thing tells a cloud what shape to adopt. The shape a cloud, a drop or a rock takes depends mostly on the set of interactions between its parts, and also on the interactions between other parts in the environment.

Therefore, the shape of a wave is not determined by the universe (by the supra-system), but by its sub-systems (the particles interacting). Interactions produce behaviors, according to our perception. The parts (assuming that such is a valid truth) determine the features of the whole, and not conversely. Considering the conditions described above, that's your answer.

Note: remark that I've considered the atom as the smallest thing. But that's not a final truth. It was an empirical truth (absolutely subjective to humans) until quarks were "discovered", and quarks were an empirical truth until strings were theorized, and such pattern continues. But we don't know the final truth. We're far from it. The empirical truth that we know is the only one we have and we're constrained to develop knowledge (including scientific knowledge) upon it. The problem of our empirical truth is that it can take the shape of a set of tautologies. Sadly, we need to accept that. We born, and learn to differentiate things based on what our mind and body determines to be things. The structure of reason has this shape and there's nothing we can do about it. We're constrained to learn from nature behind the biased spectacles of perception.

[1] https://en.wikipedia.org/wiki/Kant%27s_antinomies#The_first_antinomy_(of_space_and_time)

It depends on which wave we are considering. In the case of a wave on a string, the wave travels from the region of the string at a greater tension to the region at a lesser tension. For a sound wave, the wave travels from a region at a higher pressure to that at a lower pressure. In general, in any wave, there is a quantity that determines the direction in which the wave travels. The wave travels from a region at which the magnitude of that quantity is higher to a region at which the magnitude of the quantity is lower.

I used to wonder about this question. I am going to answer this question as follows:

Imagine a moving particle as $s( t) = t^2 $ where $s$ is the distance from some reference point ( origin) and t is the time interval passed when the particle was at the origin. now suppose we observe this particle for time $t in [2,10] $ (you can take thing is SI unit but it doesn't matter). You can calculate the velocity, acceleration at every Time in the interval.

Now we are going to reduce the interval of our observation take $ [4,6] $ then [4.5,5.5] and so on. For each interval, you can calculate velocity and acceleration at every time in the interval until you take a point. This is the same as taking a photo of the particle at a particular time.

Till now I just framed your question in a different way.

Now we take time inter as a point i.e. $tin {5}$ We can't calculate velocity and acceleration from our observation.

The conclusion is that when we take photos of a wave or a particle we have less information than taking a small clip(video). This is about the information we get from the universe.

Above I talked about the information we get by observing the universe. Now I am going to talk about the information in the universe. I am not an expert philosopher still what I think is that we can't take a point in time. We see in some sci-fi movies that time is frozen and then they replay the time and everything is normal. I believe when we replay the time. The universe not know " what to do next".

The wave equation is a second order partial differential equation so it requires both position and the time derivative of position (speed) as initial conditions. The sign of the speed gives the direction that the wave is moving--so a single photo would not be enough. (Note that the momentum can be derived from the signed speed.)