1. There is often a tradeoff between the advisor from whom you will learn the most vs the one who will help your career the most. Letters of recommendation are the most important factor in obtaining a postdoc/faculty job, and some professors are 10x as influential as others. However, the influential prof might be a jerk and not good at training students. The kind mentor with deep knowledge or the approachable junior faculty member might not be a mover and shaker.
2. Most grad students fail to grasp the big picture in their field and get too caught up in their narrowly defined dissertation project.
3. Benchmark yourself against senior scholars at a similar stage in their (earlier) careers. What should you have accomplished / mastered as a grad student or postdoc in order to keep pace with your benchmark?
4. Take the opportunity to interact with visitors and speakers. Don't assume that because you are a student they'll be uninterested in intellectual exchange with you. Even established scholars are pleased to be asked interesting questions by intelligent grad students. If you get to the stage where the local professors think you are really good, i.e., they sort of think of you as a peer intellect or colleague, you might get invited along to dinner with the speaker!
5. Understand the trends and bandwagons in your field. Most people cannot survive on the job market without chasing trends at least a little bit. But always save some brainpower for thinking about the big questions that most interest you.
6. Work your ass off. If you outwork the other guy by 10%, the compound effect over time could accumulate into a qualitative difference in capability or depth of knowledge.
7. Don't be afraid to seek out professors with questions. Occasionally you will get a gem of an explanation. Most things, even the most conceptually challenging, can be explained in a very clear and concise way after enough thought. A real expert in the field will have accumulated many such explanations, which are priceless.
Please feel free to contribute your own advice!