Do not apply to safety schools if you want an academic career.
Within any given field there are only about 5-50 top programs in the United States, and if you aren't sure where those are for your field, your undergraduate faculty can tell you. These are the only programs you should apply to. There is a huge oversupply of Ph.Ds in almost every field, and consequently it's a buyers market. Colleges hire fulltime faculty almost exclusively from a small percentage of the Ph.D programs. If you can't get into one of those top programs, it's better to reconsider your career choices now before you've invested several years of your life, than five years down the road when you have a degree that actually decreases your attractiveness to employers.
If you go to one of those top programs you have a much better chance of landing a tenure-track academic job at the other end. If you matriculate from a second rank program, you might be able to land a job at a third-rank or worse institution, if you're lucky. If you go to a third rank institution for your Ph.D, get ready to adjunct at a community college.
You may well have gone to a better high school than grammar school. You almost certainly went to a better college than high school. The move from college to graduate school is your last chance to move up the ladder of academic rankings. Doing well as an undergraduate at a lower-tier institution can still get you into a top program like MIT or the University of Chicago. But doing well as a newly minted PhD at a lower-tier institution will not even allow you to be considered for a faculty position at a top institution.
At my university, we have several professors who went here for their graduate studies. Maybe this circumstance is an exception.
There was some support for the idea of starting an advice repository for grad students much in the same tradition as the Boring Advice Repository and the Solved Problems Repository started earlier by Qiaochu_Yuan. So here goes.
Please share any advice, boring or otherwise, for succeeding at grad school. I realize that succeeding might mean different things to different people, but I believe most people largely agree with what it means in this context. Feel free to elaborate on what you believe it should mean, if you have views on the subject.
I am a theoretical physics grad student, so I'm personally more interested in advice for mathy disciplines (i.e. physics, math, CS), and I also suspect that there are many grad students from these disciplines on LessWrong; but advice for any discipline is welcome as well.
Advice is welcome from anyone, but please do mention your background for providing the advice so that people can weight the advice accordingly. For example, I would be more be open to listening to advice from someone who has completed a very successful PhD, than from someone who has simply interacted with a lot of grad students but has never been to grad school.
Also, feel free to link to advice from other sources, and maybe quote the most useful parts in what you read. Remember, this is meant to be a repository, so that people can come and find the advice, so don't worry if it seems to be something most people would've already read or known.
Thanks!