(written with Geology in mind, but probably applicable elsewhere too)
Get your Ph.D. from one of the top institutions in the field, and specifically from one of the top programs within your specialty. For most geology positions, search committees will have dozens of applicants, and they'll want to hire the best-trained person. They'll assume that a paleontologist with a Ph.D. from the University of Chicago is better trained than a paleontologist from the University of Maine, or that a sandstone petrologist from the University of Texas is better trained than a sandstone petrologist from the University of Chicago. Choosing a program on the basis of geography ("I have family in Tupleo, so I decided to get my Ph.D. at the University of Mississippi" or "I don't like cold weather, so I won't go to MIT") is likely to mean throwing away several years of your life on a meaningless degree.
See the Advice on . . . choosing graduate schools page for advice about choosing the grad school for your interests. You can look for published rankings of graduate programs, but bear in mind that all such rankings are out-of-date, in that they are the results of surveys of professionals' impressions of programs, which inevitably are based on past experiences, commonly from twenty or thirty years ago.
Work with an advisor who is well-known and well-respected in his or her field. This in part follows from #1, but remember that you will be viewed as a product of your advisor, and your advisor's letter of recommendation will be among the most important factors in the success of your job search.
Remember that everything you do in grad school (and beyond) will become material to be mentioned in letters of recommendation. Most letter-writers won't say negative things about you in letters of recommendation, but they can't be expected to invent positive things. You have to give them positive material about which to write. Some of the following (certainly the next item) provide fodder for this item.
Make yourself a positive contributor to your Ph.D.-granting department and to your advisor's research program. Solve problems rather than create problems, and look for new and better ways to get things done in your Department's teaching mission, in your advisor's lab, and so on. Don't charge in and re-organize the building or the lab, but ask questions like "I've noticed we might be able to do X better by doing it this way - would it be OK if I set things up to try it that way for a while?". You'll have at least tried to make a positive contribution, you'll have avoided unknowingly making changes in some treasured Department or lab tradition, and you'll have shown the kind of positive initiative that faculty can write about in letters.
Make sure that your Ph.D. research is an innovative way to do something that the field needs done to advance its knowledge. If you come out of your Ph.D. program able to say "I'm a person with a brand-new tool that helps us answer questions about one of the burning issues of the day", that will help you a lot more than "I'm familiar with all the techniques that were used twenty years ago, and thus I can do the same sort of work that your retiring faculty members can do and used to do".
Make sure that your Ph.D. research is what you want to be known for. Your Ph.D. dissertation will brand you as a particular kind of researcher: an expert on X but not on Y or Z. You'll have trouble getting funding thereafter to do other kinds of research, and you'll not be recognized as an important person in other areas of research for a long time, despite publishing in those areas.
Get funding for your Ph.D. project. If you're working on something for which your advisor has obtained funding, then this isn't an issue. Otherwise, you're going to have to write proposals to get money. At the M.S. level, you may have gotten familiar with proposals to GSA and Sigma Xi, but at the Ph.D. level you should be thinking about writing a proposal to major funding sources like NSF, NOAA, and the like. You'll have to be a ghost-writer, because academic institutions will only let faculty members be principal investigators of grants. However, if you can write a proposal that brings in big bucks to support your research, including your assistantship, you'll have marked yourself as a very hot commodity, because academic institutions want people who bring in money.
Publish, preferably in more prestigious journals. Applicants for faculty positions are expected to already have publications on their vitas. The people being hired today have publication records that would have gotten an assistant professor promoted and tenured twenty years ago. If you don't have that publication record, some other young Ph.D. will, and he or she will get the job.
Go to national meetings all through your grad school (and post-doc) career. It's essential that you get to know the major players in your field, and that they get to know who you are. That happens by presenting at meetings and by getting introduced, or introducing yourself, to them. You'll also learn a lot about other people's research at these meetings, and so you'll know where your field's going with its research. As a result, you'll be one of the field's sophisticates, rather than one of the people at its fringe.
Get to know as many people in your field as you can. Getting things done in science requires collaboration and exchange of ideas, and you can't collaborate or exchange ideas with people you don't know. Getting things done in science also requires getting proposals funded, manuscripts accepted, and recommendations written, and all of those things are facilitated by knowing people.
Expect that you'll have to do at least one stint as a post-doc before you land a position as an assistant professor. Increasingly, applicant pools for academic positions contain people who have done post-doctoral work. They're people who have thus worked in another lab beyond the one where they got their Ph.D. and thus have more experience. They also probably have more publications because they're had more time to produce them. Thus search committees are generally going to view candidates with post-doc experience more favorably.
Expect that you'll probably be in your mid-thirties before you land a position as an assistant professor. A typical trajectory might be age 22 after your B.S., 25 after your M.S., 28 after perhaps a little work in industry or in a lab, 33 after your Ph.D., 35 after a post-doc. This means that you're likely to be 40 or so before you get tenure and achieve whatever job security tenure provides (if tenure still exists by then).
Expect that the main focus of your career will be writing grant proposals. Yes, you'll be expected to teach. Yes, you'll be expected to do research and publish its results in the most prominent journals with the highest impact ratings. However, in the modern academic world where universities get a large part of their funding from overhead ("facilities and administration") money in federal grants, the emphasis from administrators evaluating your performance will be on how much funding you have brought in for them to spend.
All of the above is not meant to be depressing. It's merely meant to let you know some of the things you need to do, and the ways you need to approach your career, if you want a reasonable chance at an academic position. If you find the above depressing, it may be a sign that an academic career in the modern world is not for you. If you look at the above and say "OK, let's get started", then that's a good sign about your chances for a successful career.
A series of pages on "How to be a good graduate student".
Two allegedly contrasting pages from a Berkeley debate between allegedly cynical and acynical views of how to be a grad student. (Both are good).
A page of advice for graduate students from a chemist's perspective.
A page of links to pages of advice for graduate students, including some of the links here.
A short page of dissertation advice (good advice for theses too).
A series page of advice for graduate research written for MIT artificial intelligence students but potentially useful for all.
A Railsback page with advice to students about graduate school.
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