What is the relationship between graduate coursework and research? I feel like after graduate students pass their qualifying exams, they still have a lot of learning to do before they're knowledgeable enough in a specific area to understand and work on the open questions in that area. But how do they get there? Do they just take advanced courses in the area of interest, or are they expected to read up on their research area outside of coursework? If the second, should they be doing this while they're still taking classes, or do they start doing this after they're done with their coursework?
It depends. My PhD supervisor got me involved from Day 1 in research projects in his lab. He was down on programs where things were much more serial (do the required classes, pass the qual, do a proposal, do the research, defend). His students all had some research tasks from the beginning. If the student was unprepared for the task, there were papers to read or book suggestions, etc. This worked really well for me, though it may have resulted in me being somewhat under-prepared for the quals the first time I took them! I eventually passed and graduated. 

Some advisors and programs take a different approach. They really want the student to concentrate on nothing but getting through courses and the quals until they pass the quals. Then it's the proposal (if they require one). How one gets the knowledge to get from proposal to defense is up to the student and the advisor to work out. I didn't complete my coursework requirement until late in my program, so I used courses to close knowledge gaps where a relevant course was available. Where it wasn't, I worked through books and papers that I could find that helped.