It's very exciting when you can use the theory to solve "lower level" problems. For example, I'm looking forward to understanding why the quintic equation is not solvable. In the undergraduate curriculum, that seems to happen very late: only if you decide to take a class in "galois theory".
My experience with my first abstract algebra class is that it just develops theory ("why can such a group not exist? what properties must this structure have?", etc). I want to be able to see how the theory helps us in the lower level sense. Are there "example books" with interesting "lower level" problems?