Which aspects of QM are quantized? In an introductory QM class we are learning that some fundamental things in nature are quantized. For example, the energy states of a bound particle. But we have also learned that an unrestrained particle can have a continuum of energies. 
Which things in nature are quantized? Energy is, but only under certain conditions? Are position and momentum quantized, or do they have discrete quantities we just can't measure exactly? Or is it that at some level all things are discrete?  
 A: The answer depends on the system you are interested in.
For example, if the position of a particle is represented by a delta function ( or several delta functions) we can see that position has discrete spectra.
But if you look at the momentum of the particle you will see a continuous function. Not a discrete one. 
This happens because the position and momentum don't commute with each other. Because of that, there is always an uncertainty between them.
Similary, at some instances energy, will have discrete spectra ( In bound states - classically allowable regions).
If position spectra are discrete momentum spectra will be continuous and vice versa.
But you can't generalize similar relationship between energy with position or momentum. 
A: For a really thorough answer to this have a look at Reason for the discreteness arising in quantum mechanics?, though this may be a little too high level for your purposes.
As a rough guide, a system has discrete energy levels if is confined i.e. subject to some external restriction that pins it down to some region in space. For example the electron in a hydrogen atom is pulled towards the nucleus by the electrostatic force, and the result is that its energy levels are quantised. A discussion of this is given in Electron shells in atoms: What causes them to exist as they do?.
A: I stumbled across Dr. Carl Bender's lectures on mathematical methods on YouTube. He is an excellent lecturer, in his second lecture he shows how perturbation theory can be used to demonstrate  that the  eigenvalues to a perturbed hamiltonian are  smooth and continuous when the pertubation parameter is  allowed to vary in the complex plane. After deriving this result he shouts out, "quantum mechanics is not quantized!!"  I think you'll enjoy that lecture. Here is the link: https://www.youtube.com/shared?ci=P4EpkT9Ux3g . The portion relevant to your question begins around 68 minutes in. Watching the first 45 minutes of the next lecture (#3) will provide further intuition. Dr. Bender derives a logical mechanism for working out the square root of a complex number: express it in polar coordinates, square root the modulus, and divide theta by 2.  He then examines the problem with this complex square root function, mainly that the square root in polar coordinates of z=1 (e^i0) does not equal the square root of z=1 when expressed as e^(i2pi).  The solution to the problem is to introduce 'sheets' , where you consider the point at 2pi to live on a different complex plan than the point at theta=0. He then ties in quantization as the real solutions you get on these different sheets, (called Riemann surfaces) . I think Dr. Bender's  lectures 2-3 will provide exactly the intuition you are looking for.
