I kindly ask about some references concerning the representation theory of the Langlands dual of a compact Lie group, and how it relates to things related to the original compact Lie group.

My background: I know some basic facts about Lie groups/algebras, such as their root systems, Weyl groups etc. I am not familiar yet with the Langlands program, and related things. I am mostly interested for now in working over $\mathbb{R}$ and $\mathbb{C}$.

Edit 1: after some research, I realize that what I want is a reference on the Langlands classification, done by Langlands himself. So I will start by reading that article by Langlands ("On the classification of irreducible representations of real algebraic groups").

Edit 2: I found some introductory notes on endoscopy by J-P Labesse, which look very promising to me! http://www.math.utah.edu/~ptrapa/src2006/labesse.pdf

Edit 3: Knapp's article, suggested by Desiderius Severus, is indeed a really good introduction to the Langlands classification, which is part of the local Langlands program in the Archimedean case. In some of my comments, one can see that I was confused between the Langlands classification, and the geometric Satake isomorphism, which plays a role in the Geometric Langlands program. I apologize for this confusion. It took me some time to get used to some of the jargon of the Langlands program (and even now, I cannot claim to have mastered the jargon, but I have improved a little).