Picking a supervisor
Check out this other section, where I detail the process for an undergraduate looking for research experience, including how to contact professors. The process is much the same for finding potential PhD supervisors!
I do want to stress though: don’t rely too much on program rankings.
It can be tempting and I know it’s easier to simply outsource all of the work, but you rarely know the methodologies these companies use. If you do, then you already know that they prioritise all of the wrong things and where they do attempt to measure important things, they measure them using quite inappropriate metrics. Rankings are a numbers game and it’s both common and easy for universities to inflate their rankings without making any change to their research profile or internal processes. It is also the most broad view possible and does not accurately capture any useable unformation about individual departments or researchers. If you rely on rankings you can easily find yourself in a place that is wholly inappropriate given your interests and goals.
In many cases it will either be required or beneficial, although there are some notable exceptions.
In the US you would typically spend the first ~2 years of a PhD doing coursework and short projects with multiple supervisors. You may not settle to one supervisor until the end of the 2nd year and so there’s no great rush to identify that one person prior to applying. It can still be beneficial to identify a handful of potential supervisors and ask them about opportunities for 2 years’ time, just in case someone is planning to retire or move soon, but it’s more for your benefit and may not factor into admissions at all.
In some cases, contacting supervisors is actively discouraged and it should say as much on the program/department pages. In this case you should not reach out. Not following this direction is a sign that you either haven’t done your homework (you haven’t read the department pages) or you’re unable to follow basic directions. Neither is a good look.
In nearly all other cases, and particularly for programs outside of the US, reaching out to potential supervisors lies somewhere between “important” and “necessary”. Potential supervisors may play a direct role in admissions, or you may need to name a supervisor in your application along with a confirmation that they would mentor you should you be accepted. And it’s generally a good idea to get to know the person who might become your boss, and to make sure that your research interests and working styles will align.
You should, however, make sure that you actually have something to say. If your only reason to reach out is to get your name in their head and introduce yourself, don’t waste your or their time. You should also avoid asking any questions that can be answered online, by the department secretary, or which don’t yet have an answer (e.g. “are you planning to take students in 2 years’ time?"). And if you’re going to ask to set up a time to chat more about research then you ought to actually have things you want to ask or talk about.
By far the best option is by email!
You can also typically find office phone numbers online, but think carefully before you pick up the phone. Phone calls are inherently imposing, and by calling someone you are pressuring them to answer you at the exact moment you call because they don’t know who you are or why you’re calling. Phone calls and video calls are great if you’ve already agreed with the person to do just that, but I think cold calling (i.e. calling someone without them knowing ahead of time) is disrespectful. You’re implicitly stating that whatever you want to discuss is absolutely more important than anything else they’re doing, including teaching, mentoring students, holding meetings and interviews, or doing their own research. Calls should be reserved for things that are urgent and require an immediate reply. Your email about graduate positions is neither. It’s also a kinder option for those academics who work flexible hours that don’t conform to the standard 9-5, most often because they have dependents (e.g. young children) or they’re travelling.
If you email then you give the person time to think over your proposal and craft a reply. Cold call them and the answer may be a flat “no” when an email would garner a “yes”.
A general template could follow as below. (Numbers in brackets are for accessibility, and are not included in the actual email)
Salutation: see here for tips
Introduction: your name and your student status
Purpose: why you’re emailing them
Funding (Optional): if you have funding secured or have plans to apply to something big
Experience: your skills and research experience
Research interests: what you want to research in your PhD
Doing your homework: link to their work (you must have actually read these!)
Actionable request: ask a specific question or action for them to undertake
Closing: again, see here
 Dear Prof. Reynolds,
 My name is Payton Rodman, and I am currently an astrophysics Honours student at the University of Tasmania in Australia, graduating in November of this year.  I am looking to apply to doctoral programs in astrophysics, including Cambridge for the Michaelmas term in 2019.  For funding, I’m planning to apply for the Gates-Cambridge scholarship, along with the Commonwealth scholarships.
 I have research experience in theoretical astrophysics starting in 2015, mostly in the field of AGN with Radio Galaxy Zoo (RGZ), including one first-author paper currently under review with MNRAS. For my Honours thesis this year, I am investigating how Faraday Rotation of AGN jets can be used to constrain galaxy cluster properties through analytic and numerical modelling, for use with large radio surveys like the SKA.  Your work has cropped up many times in relation to that, most recently your 2015 paper on the transport of g-modes into the ICM (abs/1511.03271).
 Continuing into graduate studies, I am looking to do more with turbulence and instabilities within accretion disks around AGN and how these might lead to X-ray variability,  similar to the work you’ve done with Dr Drew Hogg (e.g. abs/1801.05836).
 Please let me know if you plan to accept any graduate students in 2019, and if you’d consider supervising me for a PhD in a related area.  I have attached my CV with more details on my research experience, and I look forward to hearing from you and discussing any options further.