You might get more answers in /r/gradschool or /r/askacademia, but I'll go ahead and give you my experience, which is that it varies immensely from professor to professor. You may find yourself immersed in reading, as you assume, taking notes on the reading and finding interesting connections with your professor's reading. You may be relegated to formatting footnotes. You might be doing proof-reading. You will almost certainly spend some time making photocopies. It very much depends on the nature of your professor's research, their personality, and the stage their project is at.