Advice from the Wise 

Steven Weinberg's Four golden lessons. 

A classic pertinent to every station of scientific life. 

Feynman's letter to former student Koichi Mano. 

In addition to observing that "No problem is too small or too trivial if we can really do something about it", he offers this balm: "Know your place in the world and evaluate yourself fairly, not in terms of your naïve ideals of your own youth, nor in terms of what you erroneously imagine your teacher’s ideals are."

Frank Timmes' teaching statement [PDF].

Clear insights into what makes an effective environment for learning.

G. V. Pavan Kumar's gems.

Best practices for both graduate students and advisors.

Daniel Green's career advice.

On job applications, starting out in research, and undergraduate research.

Richard Hamming's You and Your Research.

Another classic. Most of it applies to researchers somewhat senior than grad students. But there's a universal take-home: "Knowledge and productivity are like compound interest. Given two people of approximately the same ability and one person who works ten percent more than the other, the latter will more than twice outproduce the former. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity - it is very much like compound interest. I don't want to give you a rate, but it is a very high rate. Given two people with exactly the same ability, the one person who manages day in and day out to get in one more hour of thinking will be tremendously more productive over a lifetime."

Michael Nielsen's Principles of Effective Research.

Martin A. Schwartz's The importance of stupidity in scientific research.

Ben Barres' How to Pick a Graduate Advisor [PDF].

Daniel Schroeder on getting letters of recommendation.

My tips to junior researchers.

Peter Medawar's Advice to a Young Scientist [PDF]. 

(I haven't read this yet.)

Chris Quigg's talk and article on Nature's Greatest Puzzles. 

"Balance grandeur and sweep of the Great Questions with our prospects for answering them. Unimagined progress can flow from small questions."

Uri Alon's How to Choose a Good Scientific Problem and a companion piece, Michael A. Fischbach's Problem choice and decision trees in science and engineering. 

David Tong's How to Make Sure Your Talk Doesn't Suck.

Chris Quigg's Advice to a Lecturer.

Michael Polanyi's Republic of Science.

A primer on the sociology of what we do.