Quantum made easy: so what *is* quantum mechanics all about?

You're asking a tough question! I try to explain QM to nonphysicists all the time, and it's really hard; here are a few of the accessible explanations I've found.

What I'll say below is not logically airtight and is even circular. This is actually a good thing: motivation for a physical theory must be circular, because the real justification is from experiment. If we could prove a theory right by just talking about it, we wouldn't have to do experiments. Similarly, you can never prove to a QM skeptic that QM must be right, you can only show them how QM is the simplest way to explain the data.

Quantization $\rightarrow$ Superposition

If I had to pick one thing that led to as much of QM as possible, I'd say quantum particles can exist in a superposition of the configurations that classical particles can have. But this is a profoundly unintuitive statement.

It's easier to start with photons, because they correspond classically to a field. It's intuitive that fields can superpose; for example the sound waves of two musicians playing at once superpose, adding onto each other. When a light wave hits a partially transparent mirror, it turns into a superposition of a transmitted and reflected wave, each with half the energy.

We know experimentally that light comes in little bullets called photons. So what is the state of a photon after hitting a partially transparent mirror? There are a few possibilities.

1. The photon is split in half, with one half transmitted. This is wrong, because we observe all photons to have energy $E = hf$, never $E = hf/2$.
2. The photon goes one way or the other. This is wrong, because we can recombine the two beams into one with a second partially transparent mirror (the would-be second beam destructively interferes). This could not happen if the photon merely bounced probabilistically; you would always get two beams.
3. The photon goes one way or the other, but it interferes with other photons somehow. This is wrong because the effect above persists even with one photon in the apparatus at a time.
4. The photon is something else entirely: in a superposition of the two states. It's like how you can have a superposition of waves, and it is not a logical "AND" or "OR".

Possibility (4) is the simplest that explains the data. That is, experiment tells us that we should allow particles to be in superpositions of states which are classically incompatible. Then you can extend the same reasoning to "matter" particles like electrons, by the logic that everything in the universe should play by the same rules. (Of course such arguments are much easier in hindsight and the real justification is decades of experimental confirmation.)

Superposition $\rightarrow$ Probability

Allowing superposition quickly brings in probability. Suppose we measure the position of the photon after it hits one of these partially transparent mirrors. Its state is a superposition of the two possibilities, but you only ever see one or the other -- so which you see must be probabilistically determined.

This is not a proof. It just shows that probabilistic measurement is the simplest way to explain what's going on. You can get rid of the probability with alternative formulations of QM, where the photon has an extra tag on it called a hidden variable telling it where it "really" is, but you really have to work at it. Such formulations are universally more complicated.

Superposition $\rightarrow$ Entanglement

This is an easy one. Just consider two particles, which can each have either spin up $|\uparrow \rangle$ or spin down $|\downarrow\rangle$ classically. Then the joint state of the particles, classically, is $|\uparrow \uparrow \rangle$, $|\uparrow \downarrow \rangle$, $|\downarrow \uparrow \rangle$, or $|\downarrow \downarrow \rangle$. By the superposition principle, the quantum state may be a superposition of these four states. But this immediately allows entanglement; for instance the state $$|\uparrow \uparrow \rangle + |\downarrow \downarrow \rangle$$ is entangled. The state of each individual particle is not defined, yet measurements of the two are correlated.

Complex Numbers

This is easier if you're speaking to an engineer. When we deal with classical waves, it's very convenient to "complexify", turning $\cos(\omega t)$ to $e^{i \omega t}$, and using tools like the complex Fourier transform. Both here and in QM, the complex phase is just a "clock" that keeps track of the wave's phase. It's trivial to rewrite QM without complex numbers, by just expanding all of them as two real numbers, as argued here. Interference does not require complex numbers, but it's most conveniently expressed with them.

Wave Mechanics

The second crucial fact about quantum mechanics is momentum is the gradient of phase, which by special relativity means that energy is the rate of change of phase. This can be motivated by classical mechanics, appearing explicitly in the Hamilton-Jacobi equation, but I don't know how to motivate it without any math. In any case, these give you the de Broglie relations $$E = \hbar \omega, \quad p = \hbar k$$ which are all we'll need below. This postulate plus the superposition principle gives all of basic QM.

Superposition $\rightarrow$ Uncertainty

The uncertainty principle arises because some sets of questions can't have definite answers at once. This arises even classically. For instance, the questions "are you moving north or east" and "are you moving northeast or southeast" do not have definite answers. If you were moving northeast, then your velocity is a superposition of north and east, so the first question doesn't have a definite answer. I go into more detail about this here.

The Heisenberg uncertainty principle is the specific application of this to position and momentum, and it follows because states with definite position (i.e. those that answer the question "are you here or there") are not states of definite momentum ("are you moving left or right"). As we said above, momentum is associated with the gradient of the wavefunction's phase; when a particle moves, it "corkscrews" along the direction of its phase like a rotating barber pole. So the position-definite states look like spikes, while the momentum-definite states look like infinite corkscrews. You simply can't be both.

Long ago, this was understood using the idea that "measurement disturbs the system". The point is that in classical mechanics, you can always make harmless measurements by measuring more gently. But in quantum mechanics, there really is a minimum scale; you can't measure with light "more gently" because you can't get light less intense than single photons. Then you can show that a position measurement with a photon inevitably changes the momentum by enough to preserve the Heisenberg uncertainty principle. However, I don't like this argument because the uncertainty is really inherent to the states themselves, not to the way we measure them. As long as QM holds, it cannot be improved by better measurement technology.

Superposition $\rightarrow$ Quantization

Quantization is easy to understand for classical sound waves: a plucked string can only make discrete frequencies. It is not a property of QM, but rather a property of all confined waves.

If you accept that the quantum state is a superposition of classical position states and is hence described by a wavefunction $\psi(\mathbf{x})$, and that this wavefunction obeys a reasonable equation, then it's inevitable that you get discrete "frequencies" for the same reason: you need to fit an integer number of oscillations on your string (or around the atom, etc.). This yields discrete energies by $E = \hbar \omega$.

From here follows the quantization of atomic energy levels, the lack of quantization of energy levels for a free particle, as well as the quantization of particle number in quantum field theory that allow us to talk about particles at all. That takes us full circle.

I'm no physics grad student, but I've had to come to grips with similar oddities in QM just from my own hobby perspective. If I were to explain QM to a complete newcommer, to try to get them comfortable with the quirks, I'd start with what is probably the single most important aspect of QM. Like all other scientific theories, it is a model. It is not reality, it is a description of reality. It's just a description that's so good that it can be difficult to tell the difference.

If they can grasp that, the next step is to point out that the observations which lead to the development of QM are simply pesky. They refuse to fit together in any simple solution. The whole reason we have QM is because we started exploring corner cases, and our existing models fell apart. So when the QM models say something counterintuitive (read: entanglement), it's because there were some really pesky experiments done which showed that yes, this is the way our universe really works (Such as those demonstrating the Bell inequalities)

At this point, I consider wave-particle duality to be an essential next step. You mention not wanting to dig into it, but when explained properly, I think it adds to clarity rather than murkiness. The key is to explain that there used to be two models for how light worked. One used math associated with particles, and one used math associated with waves. We often say "light is a wave and a particle at the same time," but that's the confusing phrase. I prefer to say "light is neither an EM wave nor a particle. It is something which sometimes behaves really similar to a wave, or really similar to a particle, but when we put it into exotic situations, it behaves as something completely different. (That "something completely different" happens to be well-modeled using superposition)

At that point, if they are comfortable with this slight twist to normal terminology, you can start going into the findings and the thought experiments and so forth. I find it is far easier to explain Schrodinger's cat as "the cat is neither alive nor dead, it's something else" than it is to say "it is alive and dead," which is designed to fly in the face of their understanding.

In the end, what makes QM hard is not the QM. As you said, QM is easy. What makes it hard is that the models QM use suggest that a lot of things we take for granted on a daily basis aren't quite accurate. If one is gentle with how one unsettles all of these assumptions that have been made over years, it's much easier to get someone to be comfortable with QM.

Consider this: using the classical model of the atom, your hand and the desk are well over 99.99% empty space. It's merely electrostatic repulsions that give the perception of these objects being solid. You and I know this. But most people are not inuitively comfortable with it. It bothers them to think that objects are that ephemeral. Now if that causes cognative dissonance in someone, is it not reasonable that they would have cognative dissonance with QM, which suggests that the concept of something being empty space isn't even really meaningful?

The theory of quantum mechanics became necessary when physical measurements could not be fitted within the mathematical frameworks of classical mechanics , electrodynamics and thermodynamics.

AFAIK three experiments were fundamentally not fitting the classical models.

1) Black body radiation

2)the photoelectric effect

3) the atomic and molecular spectra , showing discrete lines.

At a lower level, the existence of atoms themselves, electrons orbiting around a positive nucleus in no way could be explained in a stable classical model, as eventually the electrons should fall on the nucleus and become one. ( I have an answer here to this which is relevant to this answer)

The Bohr model showed the way to mathematically model 3) , Planck's assumption of quantization in photon energies in the black body solved the ultraviolet catastrophy, number 2), and the photoelectric effect was explained by the assumption of discrete energies needed from the photons to release bound electrons from surfaces.

The theory of quantum mechanics tied in a mathematical model based on specific wave differential equations and a number of axioms (postulates) , explained existing data and predict new set ups.

I am trying to say that quantum mechanics is not a brilliant mathematical model somebody invented, but a mathematical description of nature in the microcosm forced on us by data.

The probabilistic nature, was forced by the fits to the data.

I'm not interested in why we need QM, I want to know/explain what fundamental assumption is necessary to build the theory.

The fundamental assumption is that a physics theory had to describe/fit existing data, that the classical theories could not, and that it should successfully predict new setups.

The fundamental assumptions/axioms are given in the postulates of quantum mechanics, so as to pick from the solutions of the appropriate differential equations, those functions that fulfill the postulates.

All your list is the mathematical consequence of the above choice, and I do not think that it something simple to explain to non mathematically sophisticated people. Complex numbers was a separate mathematical course in my time ( 1960s)

Edit after edit of question:

what QM is about, what happens in the "quantum world", what phenomena it correctly predicts, no matter how counterintuitive to our classical minds.

To introduce somebody with minimal mathematics, to the quantum mechanical framework, I would state that "there are no absolute dynamical predictions for the behavior of matter. Quantum mechanics is all about probabilities, not certainties"

In classical mechanics there is a simple equation when a ball is thrown that describes its trajectory. In quantum mechanics the microscopic trajectory of a particle is not predictable, but probable: there are many paths that the particle could take at the nanometer and smaller scales.

I would then give them the double slit interference experiment one electron at a time. The experiment is "electron scattering off two slits with specific width and distance apart", which creates the microscopic quantum mechanical framework, and the electron is detected macroscopically at the screen The accumulation of electrons shows up that the probability distribution didsplays wave interference patterns, while the point on the screen , within nanometers, has a specific (x,y) characteristic of a classical particle footprint. This demonstrates the wave particle duality.

Once the wave nature is demonstrated, the existence of phases in the description of waves allows to explain superposition and entanglement with simple analogies to water waves and pendulums BUT holding for probability distributions.

A minimum of mathematics is necessary, but mathematics is also necessary to explain classical physics behavior too.