Okay, so I've been through some basic results on representation theory. I've gone over the proof of Burnside's $pq$ theorem using characters. I've also read though the basics of Lie groups and algebras. However, I still haven't come across a theorem or any examples which set off an "Aha!" moment in which I understand what representations really are and when their use would be appropriate.
For example, when studying group actions, in my opinion the orbit-stabilizer theorem gives me a good idea of what's going on when we study actions on finite groups - I haven't found any such analogue in representation theory.
I suppose part of the problem is that representations are useful in so many distinct ways (finite groups, Lie groups, harmonic analysis, combinatorics, etc.) that I have a hard time synthesizing a coherent picture. Anyone have any good recommendations for books/topic/theorem/examples?