Was it unusual for people in those days to learn Calculus? Could a grad student take a course in differential equations or multi-variable Calculus, or did they have to learn from journals? I am always amazed with famous people like Riemann. I tried to get a grasp of analytic number theory, but it's just too hard.
It was not unusual for a student at the University of Cambridge to learn calculus.
If you look at a Tripos paper for 1786 (reproduced on page 183 of Mathematical Recreations and Essays by W. W. Rouse Ball, available from Project Gutenberg) there were relatively simple questions on fluxions and fluents.
A wider review can be found in Mathematics in Victorian Britain by Raymond Flood, Adrian Rice, Robin Wilson, and extracts can be seen by following this link from Google Books.
The current concept of a graduate student is rather modern. Those seeking an academic career would have taken other roles such as lecturers and tutors while seeking a college or university fellowship. In the early period, most in England would have been ordained Anglican priests (as Charles Dodgson/Lewis Carroll was) and used their spare time for research.
Sources would have included journals, books and correspondence. A researcher had greater personal responsibility for their own efforts than they do now, but since less research by others existed then, it was easier to keep up-to-date.