I'm beginning to study the quantum chemistry (my background is computer science and computational mathematics) and I'm not sure if I understand well the basic concepts, like wave function and operators.

I've read questions

What is a wave function in simple language?

Use of Operators in Quantum Mechanics,

and I understood that the wave function describes all the observable quantities of a system, e.g. an electron.

But what should I do when I want to obtain some of these quantities? Let's say I want to obtain spin and momentum. If I use the linearity property of quantum operators, $\hat{O}$ being the operator, $o$ its eigenvalue and $\psi$ the wave function, like this

$$\hat{O}\psi = o\psi$$

does $o$ always equal the observable value? In other words, is it enough to use the above equation and express the $o$ variable to get an arbitrary observable value?