This is part of the ongoing series of pro tips for graduate students, check out parts one and two for the original installments.
Learn how to write papers in a very clear and simple style. Whenever you can write in plain English, skip jargon as much as possible, and make the approach you are using understandable and clear. This can (sometimes) make it harder to get your papers into journals. But simple, clear language leads to much higher use/citation of your work. Examples of great writers are: John Storey, Rob Tibshirani, Robert May, Martin Nowak, etc.
It is a great idea to start reviewing papers as a graduate student. Don’t do too many, you should focus on your research, but doing a few will give you a lot of insight into how the peer-review system works. Ask your advisor/research mentor they will generally have a review or two they could use help with. When doing reviews, keep in mind a person spent a large chunk of time working on the paper you are reviewing. Also, don’t forget to use Google.
Try to write your first paper as soon as you possibly can and try to do as much of it on your own as you can. You don’t have to wait for faculty to give you ideas, read papers and think of what you think would have been better (you might check with a faculty member first so you don’t repeat what’s done, etc.). You will learn more writing your first paper than in almost any/all classes.