Updated: Advice on choosing a grad school

I realized it's been several years since I've run a version of this, and it's the right season....

This is written on the assumption that you have already decided, after careful consideration, that you want to get an advanced degree (in physics, though much of this applies to any other science or engineering discipline).  This might mean that you are thinking about going into academia, or it might mean that you realize such a degree will help prepare you for a higher paying technical job outside academia.  Either way,  I'm not trying to argue the merits of a graduate degree.

• It's ok at the applicant stage not to know exactly what you want to do.  While some prospective grad students are completely sure of their interests, that's more the exception than the rule.  I do think it's good to have narrowed things down a bit, though.  If a school asks for your area of interest from among some palette of choices, try to pick one (rather than going with "undecided").  We all know that this represents a best estimate, not a rigid commitment.
• If you get the opportunity to visit a school, you should go.  A visit gives you a chance to see a place, get a subconscious sense of the environment (a "gut" reaction), and most importantly, an opportunity to talk to current graduate students.  Always talk to current graduate students if you get the chance - they're the ones who really know the score.  A professor should always be able to make their work sound interesting, but grad students can tell you what a place is really like.
• International students may have a very challenging time being able to visit schools in the US, between the expense (many schools can help defray costs a little but cannot afford to pay for airfare for trans-oceanic travel) and visa challenges.  Trying to arrange skype discussions with people at the school is a possibility, but that can also be challenging.  I understand that this constraint tends to push international students toward making decisions based heavily on reputation rather than up-close information.  
• Picking an advisor and thesis area are major decisions, but it's important to realize that those decisions do not define you for the whole rest of your career.  I would guess (and if someone had real numbers on this, please post a comment) that the very large majority of science and engineering PhDs end up spending most of their careers working on topics and problems distinct from their theses.  Your eventual employer is most likely going to be paying for your ability to think critically, structure big problems into manageable smaller ones, and knowing how to do research, rather than the particular detailed technical knowledge from your doctoral thesis.  A personal anecdote:  I did my graduate work on the ultralow temperature properties of amorphous insulators.  I no longer work at ultralow temperatures, and I don't study glasses either; nonetheless, I learned a huge amount in grad school about the process of research that I apply all the time.
• Always go someplace where there is more than one faculty member with whom you might want to work.  Even if you are 100% certain that you want to work with Prof. Smith, and that the feeling is mutual, you never know what could happen, in terms of money, circumstances, etc.  Moreover, in grad school you will learn a lot from your fellow students and other faculty.  An institution with many interesting things happening will be a more stimulating intellectual environment, and that's not a small issue.
• You should not go to grad school because you're not sure what else to do with yourself.  You should not go into research if you will only be satisfied by a Nobel Prize.  In both of those cases, you are likely to be unhappy during grad school.  
• I know grad student stipends are low, believe me.  However, it's a bad idea to make a grad school decision based purely on a financial difference of a few hundred or a thousand dollars a year.  Different places have vastly different costs of living - look into this.  Stanford's stipends are profoundly affected by the cost of housing near Palo Alto and are not an expression of generosity.  Pick a place for the right reasons.
• Likewise, while everyone wants a pleasant environment, picking a grad school largely based on the weather is silly.
• Pursue external fellowships if given the opportunity.  It's always nice to have your own money and not be tied strongly to the funding constraints of the faculty, if possible.  (It's been brought to my attention that at some public institutions the kind of health insurance you get can be complicated by such fellowships.  In general, I still think fellowships are very good if you can get them.)
• Be mindful of how departments and programs are run.  Is the program well organized?  What is a reasonable timetable for progress?  How are advisors selected, and when does that happen?  Who sets the stipends?  What are TA duties and expectations like?  Are there qualifying exams?  Where have graduates of that department gone after the degree?  Know what you're getting into!  Very often, information like this is available now in downloadable graduate program handbooks linked from program webpages.   
• It's fine to try to communicate with professors at all stages of the process.  We'd much rather have you ask questions than the alternative.  If you don't get a quick response to an email, it's almost certainly due to busy-ness, and not a deeply meaningful decision by the faculty member.  For a sense of perspective:  even before I was chair, I would get 50+ emails per day of various kinds not counting all the obvious spam that gets filtered. 

There is no question that far more information is now available to would-be graduate students than at any time in the past.  Use it.  Look at departmental web pages, look at individual faculty member web pages.  Make an informed decision.  Good luck!