I want to understand the expected value, variance, covariance and correlation operators, e.g. $E[x], Var[X], Cov[X,Y], Corr[X,Y]$ and how to transform them appropriately. Some mention of i.i.d.s would be good and a slow crawl up to stationary processes would help.
A lot of intermediate stuff is covered in econometrics books but nowhere can I find a simple explanation of the form:
$X_t = E_t + AE_{t-1}$
$X_{t-1} = E_{t-1} + AE_{t}$ (a first lag)
$Var [X_t] = (1+A^2)\sigma_{e^2}$
Why then is the $Cov(X_t,X_{t-1}) =A \cdot \sigma_{e^2}$
Is it because these are the shared terms?
I need something to introduce me to the operators and how one can find proofs using simple operations on these terms.
Are you interested in the mathematical background? Because then you should probably look for a book/script with the topic "introduction to probability theory". I don't know you mathematical background but if you get stuck with those kind of books you might want to look for books on "measure theory" and if the convergence arguments startle you you would have to look for books on calculus/analysis.
If you define the expected value as a Lebesgue integral over the random variable. And $COV(X,Y):=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]$
Understanding Lebesgue integrals is really worth it for probability theory. You understand what independence and correlation means on a more fundamental level. It starts to get really interesting once you get to conditional expected values and stochastic processes. The proofs for the Law of large numbers and the central limit theorem are also quite nice to have seen at some point.