EDIT 2:There is no general procedure. It is not easy to find out how many commuting observables there are.
In classical mechanics some systems are integrable. Then they have as many constants of motion as the number of degrees of freedom. For a system of N particles in d spatial dimensions times the number of degrees of freedom is Nxd. One particle and a central force is central is integrable. Integers of motion are the energy, the energy, the length of the angular momentum vector and its angle with respect to the z-axis.
In general it is quite tricky to find the constants of motion, and actually a typical system is not integrable. Often the only constant of motion is the energy. Such systems are more or less chaotic.
For quantum systems the constants of motion corresponds to hermitian operators (observables) that commute with the hamiltonian. Also here, integrable systems have as many independent observables as degrees of freedom. These observables should commute with each other. They can be quite tricky to find and might not exist, except for the hamiltonian. Typically you find these observables by finding out what symmetries the system possesses.
The observables will define the quantum numbers of the states. Thus there are as many quantum numbers as degrees of freedom for integrable systems if energy is used as one of the quantum numbers. Often, the energy quantum number is replaced by a variable that counts the number of nodes in e.g. the radial direction.
Classical non-integrable systems the motion shows varying degree of chaos. Quantum non-integrable systems have "complicated" wave functions and show energy spectra where levels seems to avoid each other in energy space.
EDIT 3: Compute all the eigenstates of the hamiltonian. Construct the projection operators for each eigenstate. They are mutually commuting. Then you have as many observables as eigenstates, but not useful ones. As indicated above, the useful observables are found by first finding the symmetries, then use group theory to find the conserved quantities.
EDIT 4: The typical procedure to fully analyze a quantum system is to find symmetries, apply/develop approximations, calculate numerically the staes or some of their properties.