If quantum particles are always waves, then what are we really measuring the position, angular momentum, spin etc. of? I've been losing sleep trying to marry, in the usual QM formulation we've been learning, what my lecturer said in our problems class a few weeks back - 'quantum particles are never particles (in the classical, everyday object sense of the word) but are instead always waves', and the fact that we're talking about angular momentum, spin, orbits etc. which imply the existence of something particle-like, in my eyes at least. I understand that wave-particle duality (using the definition of a particle stated above) is a myth. Quantum particles never behave like our classical idea of a particle; I get that, I think. They act in their own weird way. But if we define a particle as a tiny bit of matter, then electrons, protons etc., are particles.  So is the apparent particle behaviour nothing but a particular case upon measurement? An illusion due to the wave function collapse to a localized pulse shape which may appear like a pointlike particle following a trajectory if measured repeatedly over small enough time intervals? So what are we measuring the position, angular momentum, spin etc. of really? Is it that localized wave packet? Or is it indeed a particle, and we use the wave function simply (not so simply) as a mathematical tool? Or have I got the wrong idea completely? Cheers!
 A: The most profound thing about quantum mechancs is that the things that we measure are not real. What this gnomic statement means was first clearly articlulated by John  Stewart Bell  in the form of the Bell inequalities.  These inequalities show that if the things being measured have an existence beyond the measurement ---" elements of reality"---- then there are limits to how strongly certain measurements can be correlated.  Experiment has shown that these  limits are violated in the real world, so there are "correlations without correlates" as paraphrased by Mermin
A: 
If quantum particles are always waves, then what are we really measuring the position, angular momentum, spin etc. of?

There is a basic misunderstanding in the concept of "wave" in this question. The quantum mechanical waves , are called that because they are solutions of wave equations, but those solutions define probability waves not space dimension waves.
I keep pointing out that the best way to get an intuition about quantum mechanical particles and their wave functions is to study the easy to find double slit experiments one particle at a time. The plot is an older photo of the experiment in the same wiki article ,

The individual electrons leave a point on the screen which seems random, and is the footprint expected from a classical particle. The accumulation gives a probability distribution that has sinusoidal variations. That is the wave nature of the particle.
This is wrong in my opinion, "quantum particles are never particles (in the classical, everyday object sense of the word) but are instead always waves'"
the word probability is missing, they are described by probability waves, the probability of finding a quantum particle at (x,y,z,t) depends on the quantum mechanical nature, the individual particle itself is not spread out in space, as the experiment above shows.
You ask:

So what are we measuring the position, angular momentum, spin etc. of really?

For a given single interaction, the decay of a neutral for example, in a bubble chamber:


A neutral particle is produced which decays into one positive and one negative particle with a characteristic 'vee’ pattern. Measurements are required to show that this is a $K^0$. more details

The experiments show tracks of what one would expect of classical particles, the momenta can be measured, conservation of charge, angular momentum etc are on a particle basis. All the particles going through the chamber leave clear particle tracks. Where is the wave? The wave nature, i.e. the quantum mechanics, is studied by an accumulation of such events which will give information about the probability of interactions and will depend on the wave nature of the wavefunctions of the particular interaction.
A: I guess the problem you're having is what is particle, and what it means by not behaving like a particle. At least my interpretation is always to avoid using terms like "particle-wave duality" or these confusing terminologies. Instead, always consider a state as a vector and a physical quantity (operator) as a matrix would significantly help you to understand things from a computational point of view, and I guess is now quite widely accepted as a modern way to teach quantum mechanics. I really recommend you to read chap 1 of Sakurai's Modern Quantum Mechanics.
