What is Quantum Mechanics really about? This question might sound very silly, so I'm sorry if that's the case. I'll try my best to make my point clear here. Before explaining, just to make clear, I'm not confused because of the Math involved. I've started now to study Functional Analysis, but I have a reasonable background in Math. What I'm confused is what is the overall idea of Quantum Mechanics.
Before I've started the course of Physics I've always heard people saying that Quantum Mechanics was all about describing microscopic phenomena (electrons, atoms and so on) so that we are able to understand the structure of matter.
Since I've started the Physics course some years ago I took some introductory courses in modern Physics and Quantum Mechanics. In those courses the main thing that was stressed were two points:


*

*The need for Quantum Mechanics, i.e. the situations on which Classical Mechanics failed to describe phenomena and predict things, was seem mostly on experiments studying the structure of matter. In other words, the need for Quantum Mechanics was only seem when dealing with microscopic phenomena.

*The basic idea upon which Quantum Mechanics is based is the wave-particle duality. So it is seem that particles on these microscopic phenomena behave as waves. Those matter waves have a direct interpretation in terms of probability amplitudes.
In other words, those introductory courses led me to think that Quantum Mechanics is all about dealing with matter waves governed by Schrödinger's equation in order to study microscopic phenomena.
On the other hand, this semester I'm taking a more serious course on Quantum Mechanics. One of the main things that has been stressed up to now is the sharp distiction between wavefunctions and kets and also between the space of functions and the state space.
I've already asked about wavefunctions and kets here and about the space of functions and state space here. I think I got the overall idea: a ket is a state vector. That is, it is one object that encodes all the available information of a system. The main point is that the ket is not a wavefunction, although it can be related to one. In other words, we have abstracted the idea of state contained inside the wavefunction picture.
Although this is quite nice, I see now a gap between that old picture I had about Quantum Mechanics and this new one. I thought Quantum Mechanics was all about wave-particle duality and dealing with matter waves. But now, we are simply talking about abstract states of a system.
More than that, I can't see anymore the connection with the microscopic phenomena. In truth, this language of "abstract states of a system" could, IMHO, be used in Classical Mechanics as well. In other words, things seem so general, that I'm not yet being able to connect with what already learned before. In truth, if someone asked "what Quantum Mechanics is all about?" today I would be unsure of what to say.
Considering all this, my question (which I believe to be quite silly) is: what Quantum Mechanics is all about? How to bridge the gap between the abstract language of state vectors living in Hilbert Spaces and the more "intuitive" picture of wave-particle duality and microscopic phenomena?
 A: To add to ACuriousMind's answer by bringing my personal experience to the table, I too had the background mathematics for many years and understood perfectly all the algebraic machinations in many textbooks but was utterly baffled. My problem was that I was "out" of QM for a long time: I had an elementary exposure to it in undergraduate engineering which was taught in a highly outdated (approximately by eighty years) way - all about wave-particle duality and particle sometimes, wave others, never shall the twain meet and all the rest of it - so when I had to delve into QM professionally (in quantum optics) many years later, I think I was expecting it to be "kookier" than it is. This view of schizophrenic wavicles is unhelpful, as becomes clear when one thinks of the world as comprising quantum fields, the interactions between them and nothing else: you might find DanielSank's truly beautiful answer to the Physics SE question "What is the physical interpretation of second quantization?"  helpful.
Let's tease out the physics by asking again ACuriousMind's question, I'd encourage you to ask and ponder deeply on, 

" .... look at classical physics first: What is it about? ... The answer is: All of the above, but none exclusively. It's just about how stuff behaves."

When you think like this for long enough, you understand that classical mechanics is every bit as kooky as QM and that, ultimately, all that we can probe the nature of being with is experiment.
Actually, even if we altogether leave aside analogies with Hamiltonian and Lagrangian mechanics and the usual "quantization of classical theory" narrative, most of quantum mechanics looks exactly like a certain formulation of classical mechanics:


*

*There is a state that wholly defines the system both in the past since the last "measurement" and in the future until it is "measured";

*The state lives in a Hilbert space (complete inner product space) and its evolution with time is linear. 

*In an isolated, time-invariant system, the linear evolution operator must be of the form $\exp(K\,t)$ because, given reversibility assumptions (one can infer the state at any time from that at any other time), the time evolution operators must be a one-parameter group. Here you can define the time parameter to be one that adds when you compose group members: this defines time as that which makes the evolution "regular" or "even"; all other possible parameterizations are continuous bijections $\mathbb{R}\to\mathbb{R}$ of "regular", "good clock" ones (which are defined up to an affine transformation $t^\prime = A t + B;\,A,\,B\in\mathbb{R}$)).
Control and systems theory views all linear systems - classical or quantum -  in this way. The only difference is that we replace "all time" with "between measurements", the latter notion yet to be defined. The dynamics of any system described by any finite number of linear DEs of any finite order can be cast into the above form. In control theory, the time evolution operator is called the state transition matrix, whose expansion for a time varying system is given by the Peano-Baker series, which in quantum theory takes a slightly different form in the Dyson series. Witness that the above form would be something Laplace would be wholly comfortable with, with his philosophy of a clockwork universe whose behavior is defined for all time by a state at any time, until we replace "all time" with "between measurements". For how quantum mechanics differs from classical systems theory is precisely the nonunitary, measurement part. What's actually happenning at measurement is the ongoing measurement problem but what we know about is:


*

*Measurement is described by "observables", which are self adjoint operators acting on the Hilbert space of states together with a recipe for how to interpret their measurements;

*The recipe is this: firstly, straight after a measurement described by an observable $\hat{O}$ the system state is in an eigenstate of the observable $\hat{O}$;

*Secondly, the measurement is the eigenvalue corresponding to the eigenstate the measurement forces the system state into ("how" the state happens to end up in this eigenstate is the measurement problem);

*Thirdly, the choice of eigenstate in point 2. is random. The probability distribution of the measurement is wholly defined by the quantum state either as the probability of the measurement's selecting eigenstate $\psi_{\hat{O},\,j}$ square magnitude of the projection of the normalized system state onto this eigenstate or, equivalent, the $n^{th}$ moment of the probability distribution is $\mu_n=\langle \psi|\hat{O}^n|\psi\rangle$, whence one can find the characteristic function of the distribution $\langle \psi|\exp(i\,k\,\hat{O})|\psi\rangle$ ($k$ the Fourier transform variable), whence the distribution itself.
And that pretty much is it for the physical part of quantum mechanics as far as pure states are concerned. One needs to explore the notion of classical mixtures of pure quantum states through the Wigner friend thought experiement and density matrix formalisms, but these are pretty much wholly defined by the above.
The description I have given sometimes goes by the name of the "Quantum Measurement Approach". The measurement problem is an area of active foundational research, and the meaning of random is left undefined, at least until the measurement problem has an accepted solution. For now, random simply means unknowable through foreknowledge. Indeed some serious philosophers probe the nature of the as yet less than fully understood notion of randomness and chance by using quantum physics as a model for their notions. Rather than beginning with the realization that classical statistics needs broadening to the complexified quantum paradigm, they work the other way around, saying that quantum mechanics is our real, experimental world, the stuff we know and can understand directly through questioning Nature through experiment and work towards a rigorous foundation of the notion of probability as a certain, approximate abstraction of the real, "visceral" quantum mechanics we experience in the laboratory. Often students are greatly put off by the lack of applicability of classical statistics and the need for the broadened, complexified quantum statistical notions. But QM is actually the easy stuff: the stuff whose questions Nature will answer through experiment. THe notion of probability is the hard bit. See this page on the Stanford Encyclopedia of Philosphy "Chance Versus Randomness".
What I would think of as my first sound introduction to QM is the first eight chapters of the third volume of the Feynman Lectures.
So in summary:


*

*Much of quantum mechanics, aside from the measurement problem, is not too different from Laplace's conception of the World, and the unitary state description resembles the modern linear systems theoretic conception of any physical system;

*Systems theory is often linear for convenience (as an approximation to nonlinear behaviors), but several quantum mechanical notions (no cloning theorem for example) depend on the assertion that QM is exactly linear. This is one curious difference;

*Systems theory does not often deal with infinite dimensional systems. Quantum theory often lives in the complex, separable Hilbert space;

*The probabilistic nature of the measurement problem means that the norm of states has to be unity (so that the probabilities of all possible experiemental outcomes sum to one), alternatively, that states are rays through the origin or points on the unit sphere in the Hilbert space, rather than general points as in linear systems theory. Global phases applied to states have no physical meaning;

*The probabalistic measurement also implies the uncertainty principle when observables do not commute.
A: Taking a look at your questions, I would suggest you forget about the wave particle duality being this foundational principle of quantum mechanics. Rather, I would say there are two (at least, these two are the most obvious) important defining characteristics of QM, which are made clear in the Hilbert space formalism:


*

*States are represented by vectors, and while some states roughly correspond to a classical picture (such as a state with a well-defined z-component of the spin), you can also take superpositions of states, as in the Schrödinger's cat "paradox". These superpositions have no classical analog: it's as if your system was in two states at once.

*You can only calculate probabilities. If your observable $A$ can take a number of possible values ${a_n}$ and we call $|\phi_n\rangle$ the corresponding states with a well defined value of $A$, then if your system is in some state $|\psi\rangle$ the probability of measuring $a_n$ is $|\langle \phi_n | \psi \rangle |^2$. If two observables don't commute, you can't have a state where both of them have a definite value.
If you take these two principles together and the idea that a particle moving in space has a basis given by states $|x\rangle$ of definite position, then you observe interference when taking the squared modulus. This is the origin of the wave particle duality, but it's just a particular case of what the formalism lets you do.
Have you seen examples of physical systems worked out using the Hilbert space formalism? The harmonic oscillator and the hydrogen atom are the two classic examples, and they let you connect the wavefunction idea with this abstract way of describing things, but also take a look at things like spin precession in a magnetic field. The wavefunction approach won't work because if you don't care about the particle's movement there is no wavefunction. There are only complex vectors, and in the finite dimensional case using kets is essentially the same as using n-tuples of numbers, only the notation is different.
If you asked me what QM is all about I would probably mention the two principles I stated above. Entanglement and similar phenomena might also be involved, but I think this gets to the heart of the matter. It should also help you see why this formalism isn't used in classical mechanics: superposition and probabilities are not the way to do CM.
A: We don't know what quantum mechanics is about, the theory is formulated in an instrumental way. The postulates of quantum mechanics tell you how to compute the outcome of experiments. When you try to look beyond this, take into account that the experimental apparatus used, the observers etc. are  also made out of the atoms and molecules, you are forced to modify the postulates. But there is then no consensus in the physics community on how to do this.
The root cause of this is the fact that quantum mechanics has been so successful, there are no experimental results that are in conflict with it to guide us in the right direction.
A: It might help to distinguish two possible understandings of the question "what is quantum mechanics about?":


*

*What kinds of physical systems, processes, etc., is quantum mechanics particularly useful for representing; how, that is, should we use or apply quantum mechanics?

*What is the theory saying about the nature of the world; how, that is, should we understand or interpret quantum mechanics?


The second of these questions has no uncontroversial answer: issues about how quantum mechanics should be interpreted have been heavily contested since the theory's inception. The first question, on the other hand, is more easily answered - except insofar as it abuts onto the second! So in getting a handle on quantum mechanics, it's helpful to keep in mind that quantum mechanics is a theory whose application is extraordinarily well-worked out and successful, but whose interpretation remains controversial. Some people would say that this means you should resist thinking about interpretations at all, and just learn how to crank the handle to get out results and applications. I think a better approach is to learn about different interpretations, different formalisms, and different "pictures" of quantum-mechanical phenomena - but be aware of the foundational disputes, try to keep as open a mind as possible, and always be sure you know how to translate from one picture to another.
This applies, in particular, to the relationship between the wavefunction and state-vector formalisms. I would agree with you that the wavefunction formalism is somewhat more intuitive: indeed, Schrödinger felt that a key advantage of his wave mechanics was its "Anschaulichkeit", or "visualisability". Nevertheless, the state-vector formalism is generally regarded as the more fundamental. The relationship between the two formalisms is standard textbook material, but just to rehearse it quickly: the point is that wavefunctions arise as a way of representing state-vectors, if we choose a particular basis for Hilbert space. More specifically (but being somewhat sloppy about the technicalities), let $|\delta(x)\rangle$ be the state-vector representing a particle localised at the point $x$; it is an eigenvector (eigenvalue $x$) of the position operator $\hat{X}$. In certain situations, the set $\{|\delta(x)\rangle\}_{x \in X}$ forms a basis for Hilbert space (where $X$ is normal, three-dimensional, position-space). That means that given any state vector $|\psi\rangle$, we can express it as a weighted sum of elements of the position basis $\{|\delta(x)\rangle\}_{x \in X}$:
$$
|\psi \rangle = \sum_{x \in X} \psi_x |\delta(x)\rangle
$$
Here, $\psi_x$ is a complex number, the coefficient of the basis vector $|\delta(x)\rangle$. But the collection of complex coefficients $\{\psi_x\}_{x \in X}$ can equally well be thought of as a single function $\psi(x): X \to \mathbb{C}$. That function $\psi(x)$ is, of course, just the wavefunction that you're used to.
The reason to regard the state-vector formalism as more fundamental is that wavefunctions can always be regarded as ways of representing state-vectors, but not all state-vectors can be represented as wavefunctions (where I mean "wavefunction" to refer specifically to functions on position-space). For example, if you were just looking at the spin degrees of freedom of a system, then you would be dealing with state-vectors whose Hilbert space does not have the set of position eigenvectors as a basis. So the state-vector formalism is more abstract and general than the wavefunction formalism.
However, that brings us naturally to your concern that the state-vector formalism is so abstract and general, you're not sure what about it is distinctively quantum-mechanical! First, you're absolutely right that "this language of "abstract states of a system" could, IMHO, be used in Classical Mechanics as well". The difference, however, lies in the structure of the mathematical spaces we use to represent those states. The distinctive thing about quantum-mechanical state-space is that it exhibits a linear structure: there is a physically relevant notion of "adding states together" (that is, saying of a state $c$ that it is the sum of states $a$ and $b$ is a physically important claim). This is not the case in classical mechanics. This, of course, is just a way of saying that quantum mechanics - unlike classical mechanics - features superposition as a phenomenon. (Disclaimer: I don't know enough about the Koopman-von Neumann formalism to say exactly how it fits into my remarks here).
Finally, a brief remark on whether quantum mechanics pertains only to microscopic phenomena or not. This kind of has two answers, corresponding to the two questions I distinguished above:


*

*When a system has many degrees of freedom, the classical and quantum predictions for how that system will behave converge. So the systems for which one needs to use quantum mechanics to get accurate predictions are those with few degrees of freedom, which typically means microscopic systems. (This is a bit imprecise, but will do; if you're interested in the details, look up decoherence theory.)

*Whether quantum mechanics applies to macroscopic systems depends on what interpretation you favour. On "collapse" interpretations like the Copenhagen or GRW interpretations, quantum mechanics only applies to microscopic systems; on "no-collapse" interpretations like de Broglie-Bohm or Everett, quantum mechanics applies to all systems - but for reasons of decoherence, classical mechanics provides an excellent approximation when dealing with macroscopic systems.

A: Quantum mechanics, is, to the best of our knowledge, the way (almost everything in) the world works.
It's not solely about describing "matter waves", although this was fundamental to its inception. It's not solely about describing microscopic phenomena.
It's about a fundamental conception of "mechanics" (it's in the name!), an attempt to describe how physical systems behave. All physical systems. There is no border between the classical and the quantum. There is a smooth scale on which the classical approximation to quantum mechanics becomes good enough, and the quantum description hopelessly overcomplicated.
But quantum physics is not restricted to a particular type of systems (well, almost - it can't properly deal with systems where gravity should be described fully quantumly, but those situations are exceedingly rare!). Classical physics emerges from it in many senses, although you might find disagreement on how exactly it does in full generality.
And, as to all your other questions, I would ask you to look at classical physics first: What is it about? Particles moving? Waves spreading on the surface of a lake? Planets orbiting the sun? Electromagnetism? Kinetic theory, and thus thermodynamics? The answer is: All of the above, but none exclusively. It's just about how stuff behaves.
You can even describe classical physics in terms of Hilbert spaces, it's called Koopman-von Neumann mechanics. Then, both classical and quantum mechanics are unified by being described by vectors in a Hilbert space, with an algebra of observables acting upon it, and expectation values given by the Born rule. Essentially, all the difference between classical and quantum mechanics is that quantum mechanical observables don't commute, the idea that it is possible that a state has a well-defined value for $A$, but not for $B$.
A: 
The basic idea upon which Quantum Mechanics is based is the wave-particle duality.

The basic idea is that classically dynamics of configurations fails and needs to be modified. Particle wave duality is ultimately too vague to be a fundamental explanation.

So it is seem that particles on these microscopic phenomena behave as waves. Those matter waves have a direct interpretation in terms of probability amplitudes.

This is wrong. Firstly reasoning by analogy isn't precise enough to make predictions. And secondly even if you put amplitudes in front of the word probability it can make you think there is some fixed sample space and some random variables defined on it and that the components of spin for instance can be assigned to some point in the sample space. Which fundamentally contradicts the fact that noncommuting operators definitely and objectively change the state rather than reveal some preexisting property.

In other words, those introductory courses led me to think that Quantum Mechanics is all about dealing with matter waves governed by Schrödinger's equation in order to study microscopic phenomena.

That might be more right than you think. Just don't rush to think they are so tied to probability or that anything is as tied to the words we use. The theory is just what makes the predictions and how you make the predictions is the theory and the words can be misleading when they make you think something other than what the theory predicts.

In other words, we have abstracted the idea of state contained inside the wavefunction picture.

Indeed all you need is the dynamics. You can even use the Heisenberg Picture where states don't change but operators do. Or a Schrödinger Picture where the fundamental operators for a self contained system are constant but the states change.  All you need is enough for the dynamics. And a wave function is basically selecting a picture (Schrödinger Picture) as well as a basis (the position basis).

How to bridge the gap between the abstract language of state vectors living in Hilbert Spaces and the more "intuitive" picture of wave-particle duality and microscopic phenomena?

There isn't a gap. The abstract version just doesn't pick a basis. You can compute the frequency of different results by using say the energy states if the Harmonic Oscillator as your basis states or you can use square integrable wavefunctions. Ultimately you are talking about the same Hilbert Space.  Its no different than using vector calculus and then picking a basis later for a particular problem after you know what is most convenient.

Considering all this, my question (which I believe to be quite silly) is: what Quantum Mechanics is all about? 

If you want to see non relativistic quantum mechanics as something distinct from classical physics it might help to look at dBB theory. In de Broglie Bohm theory you have a wave function defined on configuration space. And the square does give a straight up probability density for the position (but don't think regular probability theory applies to things that aren't position) and the phase gives a velocity for any point there. A point in configuration space and then it is set up just like the classical Hamilton-Jacobi theory to say how the velocity changes. However with one extra potential. So you can see that a given configuration evolves one way in classical physics and that way depends solely on the classical potentials and the one configuration.
But in quantum mechanics there is an extra potential and that potential is different depending on the state. For instance in the ground state of the hydrogen atom the extra potential (called the quantum potential) produces an extra force that is exactly equal and opposite to the electrostatic attraction and the ground state has the electron at rest relative to the proton. So it just sits there. Every singe configuration of the electron proton system has a quantum potential (when in the ground state) that cancels the force from the electrostatic potential.
So you get different dynamics and the state encodes how the dynamics for a configuration deviate from the classical dynamics. That's why we wanted quantum mechanics and that is what quantum mechanics does. That is what quantum mechanics is.
But there is more, intrinsically more. The point is that we don't know what the configuration is. And here dBB can be misleading. We don't and never do know the configuration. There is a wave assigning values to lots of configurations and when one subsystem interacts with another subsystem first doesn't learn other subsystem's configuration.
We couple the states not the configurations. You (they dynamics) can branch the collections of configurations into groups that act independent of each other. And a subsystem of the configuration can be a description of a whole constellation of branches, but you never ever get the configuration. But we never found true configurations in classical physics either. No classical subsystem ever had inside it a perfect representation of the true exact state of another subsystem, not by acquiring it through an evolution of observation.
So the dynamics we ultimately track are not the configurations themselves but the descriptions of collections of them. But that is what states are.
A: When the theory of relativity was developed, Minkowski created the mathematical tool of space-time to easily describe and solve for transformations using geometric methods. Even Einstein at the time viewed this as a mathematical tool but later realized that the actual geometry of space-time was the explanation for gravity and non-Euclidean geometry of space-time is accepted as a description of our physical universe.
As an analogy, Hilbert space is a similar mathematical tool to facilitate the solution to quantum mechanical problems. Even energy is an abstact concept, but many problems could not be solved easily without the use of energy. Now whether there is a tie between Hilbert space and the physical world is a different matter. Who knows what the undelying physical reality is?
