What is the basic postulate on which QM rests. Is it that the position of a particle can only be described only in the probabilistic sense given by the state function $\psi(r)$ ? We can even go ahead and abandon the particle formalism as well. So what is the QM all about ? A probabilistic description of the physical world ? and nothing more ?
Existence of non-compatible observables: measuring one of them (say, coordinate) leads to an unavoidable uncertainty in the result of a subsequent measurement of the other (say, momentum). This is the essence of the Heisenberg uncertainty principle in the kinematics of your system. There is a detailed discussion along these lines in the beginning of the Quantum Mechanics volume (volume III) in the Course of Theoretical Physics by Landau and Lifshitz. Any measurable (physical) system, be it particle, atom or anything else, is quantum only if you can identify a manifestation of Heisenberg uncertainty principle (non-commutativity of observables).
To me, the most basic postulate is that energy comes in discrete packages of $h \nu$. Based on this assumption, you get much of the rest of basic quantum mechanics.
In fact, the Schrödinger equation is related to the Hamilton-Jacobi formulation of classical mechanics with the added assumption of quantized energy, and the Heisenberg picture follows directly from the Poisson bracket formulation of classical mechanics, assuming quantized energy.
Wave-particle duality is also a very important assumption- this gives us a way to interpret exactly what these equations express, which is the probability of locating a particle per some (small) volume of position- or momentum-space.