Once a student has gained some intuition for the basics of fluid mechanics, I think that the best and quickest path to mastery is to first learn tensor analysis and then use that knowledge to learn continuum mechanics. Better yet, take a graduate-level course on continuum mechanics and learn both at the same time. (But try not to schedule any other courses for that semester. Continuum mechanics ain't easy at first).
Sadly, I cannot recommend a text on continuum mechanics. When I took the course, I used T. J. Chung's text, which I think is fantastic. However, I suspect that I may be biased because I had the privilege of taking the course from the author himself. As a result, I had direct access to insights (as well as some god-awful homework problems) that are not available to other readers.
Regardless, my point is this. Prior to taking continuum mechanics in my 2nd year of grad school I had taken 4-5 semesters of fluid mechanics, 3-4 semesters of solid mechanics and 3 semesters of thermodynamics. Yet in one semester of continuum mechanics, I learned far more about each of those three subjects than I had learned from all of those other semesters of work combined. What I appreciated most was the mathematically concise revelation that each of those subjects are really pieces of a larger, intricately connected whole. Continuum mechanics also uncovers the natural coupling between fluid mechanics and electricity and magnetism in the form of plasma dynamics (a.k.a. magnetohydrodynamics). The underlying mathematical tool of tensor analysis is also very useful in other fields such as classical mechanics and relativity (and probably others that I'm not familiar with).
