I am seeking for a good first reference on algebraic groups, or even linear algebraic groups, where the general theory could be understood through example for the classical groups.

Understanding the theoretical definitions of tori, unipotent radical, reductive group, root system, parabolic subgroups, Levi decomposition, etc. is one thing. I always find a huge and sometimes unbridgeable gap between those definitions and formal manipulations and... what could I say for a specific example, say some classical linear algebraic groups. I found the notes of Murnaghan in the Clay Math Proceedings which are really understandable and works out lots of examples. However, in such expository notes of some pages the general theory, motivations and further examples are lacking, and I would like to find something more comprehensive to begin studying (and practicing!) algebraic groups.

So I wonder, what reference could you advise for learning algebraic group, illustrated by lots of computable examples?

Here are some precisions: I work with automorphic forms, and consider mainly adelic points of an algebraic group as well as its local components. I would like to understand how to deal with and build an intuition on what that means. For instance: what are the relations between compacity and existence of parabolic subgroups? why are the tori so central and important? what are the usual isomorphism between quotients and sequences? etc. I believe I can encapsulate it in: understanding what is exactly behind Murnaghan's notes for number theorists?