How do you start researching PhD programs?
I would talk to people who are in my field of interest. If you don't personally know such a person you can generally find one within a few degrees of separation. (For example, a work colleague of my brother has a daughter interested in astronomy and they sent her to me because I would know where to send her.) I think it is not so helpful to look at web sites or rankings. These are fickle and often out-of-date and/or misleading. Some great research groups don't maintain web sites and other research groups care way more about selling themselves than doing great work. In the end, which university you attend is not as important as the person you work with. It is your advisor and his/her support, mentorship, and contacts, that will determine your experience and your opportunities. And finding a good advisor match is best done by obtaining information from people who know that advisor—people already in the field.
What should you look for in a PhD program?
You want to go to a program that is both challenging and supportive.
You want to be surrounded by a cohort of students and faculty who are excited about research, excited about working on hard problems, and want to see everyone around them succeed at solving those hard problems.
What questions should you ask in your campus visits?
You want data, such as time to PhD and placement after PhD. These may be very different for different research groups within the same department. Senior students in the group and the director of graduate studies of the department may know this information. Before joining any research group you will want to discuss with as many students as possible what their experience has been like, both positive and negative. That is because you can read research papers to see if the research interests you, but the only way to know what it feels like to be working until 2am on something due in the morning is to ask someone who has already done so in that group.
What are some common pitfalls to avoid in the application process?
(1) Sending a generic statement without doing your research. You will want to know enough about the research done in the department to which you are applying so that you can make an argument for why they should want to admit you.
(2) Saying that you are interested in a specific group or professor without any back-up. Professors go on sabbatical, switch universities, lose funding, have too many students already, etc. If your statement indicates you have only one person in mind to work with and that person is not able/willing to take you, you won't be admitted. So be sure to be able to speak knowledgeably about more than one group.
Plus it is never a good idea to matriculate to a department that has only one group or one professor who is a good match for you. Personalities clash, life happens, and you want to go to a university where you have more than one option.