How can we know the state of a quantum system?

Experimental determination of $c_i$ values starts with preparing multiple identical systems, then making measurements. From all the measurements, one determines the probabilities, which are the $|c_i|^2$. The square root of the probabilities will tell you the $c_i$ to within a phase factor of the form $e^{i\beta}$, where $\beta$ is real, and may or may not be determinable.

A simple example of multiple identically prepared systems would be a sample of a radioactive mineral with a single parent nuclide.

Once a measurement ( observation ) is made on a quantum system the system will be in an eigenstate of that property, so if the energy of an electron is measured the electron will afterwards remain in an energy eigenstate ( until some other measurement or interaction occurs ), but if the angular momentum is measured the electron system will afterwards remain in an angular momentum eigenstate, again until some other quantum interaction occurs.

The mixed state equation $\sum_i c_i \phi_i $ is used for time forecasting of quantum systems: solve the time-dependent Schroedinger Equation, write the solution as a mixed-state equation, and then time-evolve to whatever future time you are interested it , then your statement " the probability of the eigenvalue $a_i$ associated to the eigenstate $|\phi_i\rangle$ of coming out when $A$ is measured is determined by $|c_i|^2$." is valid.