I guess I should speak regarding some portion of my independent studies: I have made what feels like a substantial amount of progress in mastering Ravi Vakil's algebraic geometry notes, in the sense that I am roughly on track to have finished a first reading by the end of the term. Also, I have solved many exercises out of chapters II and III in Hartshorne. I've learned a lot of other algebra topics too, of course...
To say something really vague: The freedom that comes with being an RA means that I can devote long periods of unscheduled time towards focused thought, towards skepticism and technical thoroughness, and towards naive curiosity and adventurous questioning.