How do we get the expectation value formula?

Observation, it fits to the measurement results (there are also some theoretical arguments that it cannot be everything, but those are details. The main point is: the theory was constructed to match observations).

Do you mean complex conjugation? It does not have to, you multiply complex numbers here.

 

There is also an analogy to probability theory, based on the Born rule that ##\psi^* \psi## is a probability distribution. The expression of the quantum-mechanical expectation value
$$
\langle \hat{A} \rangle = \int \psi^* \hat{A} \psi d\tau
$$
is analogous to the expectation value for a continuous random variable
$$
E[X] = \int_{-\infty}^{\infty} x f(x) dx
$$
especially in the case you mention where you want ##\langle x \rangle## for ##\psi(x)##.

 

You might be interested in checking out Gleason's Theorem:
http://kof.physto.se/theses/helena-master.pdf [Broken]

For the basis of Schrodinger's Equation see (as has already been mentioned) Chapter 3 - Ballentine - Quantum Mechanics - A Modern Development:
http://www-dft.ts.infn.it/~resta/fismat/ballentine.pdf

Believe it or not it actually follows from symmetry - strange but true.

Actually so does Classical Mechanics for that matter - and how it is connected to QM is very interesting - see Landau - Mechanics for example.

In fact this is one of the very deep revelations of physics and well worth becoming acquainted with:
http://www.pnas.org/content/93/25/14256.full.pdf

All books on QM suck in their own unique way and its a matter of taste what approach you take to and book you gravitate towards. I personally love Ballentine - to me it was a revelatory book on QM - but opinions vary. Check it out for yourself and make up your own mind.

Thanks
Bill