One of the best things that grad students can do is to model the behaviour of professors who they find interesting, productive, kind, supportive and so on. Figure out how those you look up to do it, and take advantage of the things they've learned through their own sweat and tears. Everyone with jobs in the academy are intelligent people. But some are exceptionally so, and some are exceptionally well-organized, disciplined, and so on. Some thinkers leave their mark on a field and cast a long shadow. These people have secrets that others don't. If they are too modest, then many students miss out on the chance to pick their brains.
Now, people will respond that students shouldn't merely go star gazing. The less popular, but still highly intelligent professors of the academy still have much to offer, and possibly more since their time is likely more their own. Success brings a lot of responsibility in terms of speaking engagements, books, articles, and so on. Some big shots are lousy supervisors. But some aren't! Those modest giants who seamlessly navigate the corridors of their schools, blending into the masses, are likely also ones who seem kind, supportive and so on. Combine that with their place in their field, and you have on your hands a great learning opportunity.
So professors, if you are hot shots, don't keep it to yourself. Be a hot shot, and show me the ropes!
I guess this goes for grad students as well. I hate the denigration that gets heaped upon ambitious students who work hard and produce presentable, publishable, works. We need more, not fewer, of these students. What we need less of are arrogant, vicious, and unsupportive students, as well as those who suck the energy from those who want to better themselves and their skills. There is a negative force at two poles: both those with too many excuses for their own ineptitude, who try to drag others down; and, those successful students who have the attitude that if you aren't on their level just now, then you are a loser.
I'd like to see more students taking seriously the careerist aspects of graduate school, while maintaining a positive, helpful, and cooperative attitude toward their peers. Seeing your name in print is a shot in the arm, which is well-deserved for surviving all the times you get rejected before you get there. Also, writing is what we do. Our ideas, as well-formed as they seem in our minds, are never as well-formed as they will be when we write them down to present to a public. We think through our writing, so more writing, is, in principle, more thinking. As a philosopher, this is a good thing.
One thing that I've come to realize is that writing begets writing. The more you give talks, publish papers, workshop, and so on, the more opportunities to do these things you will have. You make many connections in and through the activities that are a normal part of our jobs anyway. Someone will overhear the brilliant presentation you gave based on your thesis - work you have already put in - and invite you to write a chapter for their book. This really happens, and it's how some people manage to get so many publications. Journals accept an incredibly low percent of submissions (5-10%), so you will get rejected many more times than accepted. If you get invited to write or talk, then you circumvent that whole process. The lesson: write more, present more, so you can write more and present more and thus THINK more too.
I strive to achieve both the kind of attitude toward my peers that I described above and toward my work, though I am not particularly successful myself. Some of the things I do, that you can too, to be a better colleague are: make your work available to others, give them honest advice based on your own trials and pitfalls, and take their projects seriously when they have something serious to share. Build a community and you will reap the rewards.