I think when it comes to second courses, the question OF WHAT you take your representations becomes more relevant. I will say something about representations of Lie groups because that is where I ended up, but sometimes I had the feeling that people working on representations of other stuff (quantum groups, double affine Hecke algebras, braid groups, Kac-Moody algebras, Lie super algebras, etc etc) were more happy than I was. But perhaps that is only how they appear when other people are present, it is hard to tell.
Anyway, in the representation theory of real Lie groups, the thing that distinguishes the stuff for first and second courses (or put more opinion-based: distinguishing the most beautiful theory in the world from a more complicated, less intuitive and much, much more technical version of it) is a simple one: whether you work with finite dimensional or infinite dimensional representations.
When your group is compact all irreducible representations are finite dimensional and every representation decomposes as a direct sum of irreducible ones. Hence no need to look at anything infinite dimensional there. For non-compact groups however, even ones whose Lie algebra is semi-simple, none of these two statements is true anymore!
At the same time, of course, we would still like to know what are the irreducible representations and how do they combine into other, non-irreducible but still natural representations (e.g. spaces of functions on geometric objects whose symmetry group is a quotient of your Lie group) if not by simply taking their direct sum?
I think a good 'second course' type book on this kind of questions is 'Representation theory of real Lie groups, an overview based on examples' by Anthony Knapp (1986).
A specific problem (or family of problems) that is really beautiful in finite dimensions (and discussed very well in Fulton and Harris) and only partially understood (while still very beautiful) in the infinite dimensional case is the branching problem: given an irreducible representation of a big group $G$, what does it look like (i.e. what subrepresentations appear) when we we restrict the action of $G$ to a smaller subgroup $G' \subset G$?
There is really a big zoo of different types of answers to this depending on the nature of $G$ and $G'$. Someone who has written a lot about this (and really clearly at that) is Toshi Kobayashi. You can check his website to see if he has written any good introductory texts (I am too lazy to do it now) about the subject but I think the first few sections of this may qualify (the rest is probably more like 3rd-or-higher course material): https://link.springer.com/content/pdf/10.1007%2F978-3-319-23443-4_10.pdf (which is a chapter in this: https://link.springer.com/book/10.1007/978-3-319-23443-4)
