Please Note: The main points of the question below are in bold in order to minimize the time required to read the question.
Let me begin by stating that I understand representation theory is a vast and deep area with many different subfields. Of course, any learning roadmap request for representation theory would necessarily have many different answers or at least one answer with many different suggestions. I would be more interested in "mainstream topics in representation theory"; one could define this as "the set of topics which every serious representation theorist should know" (although even this is subjective and varies from subfield to subfield). Of course, I am happy for people to suggest topics which they feel are not necessarily "mainstream representation theory"; I would be interested in as many suggestions as possible.
I am interested in representation theory both as a branch of mathematics in its own right and as a set of tools and ideas which one may use to study different (either related or a priori unrelated) areas of mathematics (please feel free to interpret this in a broad sense). My background in representation theory is almost all of (and will soon be exactly) the contents of the book entitled Lie Groups by Daniel Bump. The interdisciplinary nature of representation theory dictates that I have reasonable background in other branches of mathematics; I think that I have such a background but feel free to assume as prerequisites any branch of mathematics when giving suggestions.
I am interested in studying representation theory beyond that which is covered in Daniel Bump's Lie Groups. In other words, I am happy for suggestions for topics that a potential representation theorist should know after reading Bump's book (this is the key point). Of course, I am also interested in hearing suggestions for topics that a potential representation theorist should know even if they are virtually disjoint from Bump's book. I am certainly happy for suggestions to take either the form of a textbook, research monograph, research paper, or some other form that I have not thought about.
I am not really interested in suggestions for topics that are already subsumed in Bump's book; I certainly do not object to such suggestions but they would not really be in response to this request. (You can view/download free and legally the table of contents of Bump's book at the following website: http://www.springer.com/mathematics/algebra/book/978-0-387-21154-1.)
Thank you very much for all suggestions!
Best Answer
Meta-answer: There are short introductions to a variety of interesting topics in Representation Theory of Lie Groups, a conference proceedings containing lecture notes by Atiyah, Bott, & other luminaries.