How do you start researching PhD programs?
Ask current PhD Students in your field. They have already done the work you need to do; you don't need to do it all again. Learn from their successes and mistakes.
What should you look for in a PhD program?
Look for a program that will allow you to become an independent researcher; you should also look for a place where your personality fits, and where you'll have enough money to live on (it is not as easy as you think to figure this last one out).
What questions should you ask in your campus visits?
(a) How many students have quit the program? Why did they quit? (b) Do the faculty support the efforts of graduate students toward professionalization in their discipline? How do they do this, specifically? (c) What is the actual placement record like—not just the placement record reported on the department website?
What are some common pitfalls to avoid in the application process?
(a) Don't check websites where students self-report their admissions results; you'll only drive yourself crazy. (b) Don't not have a backup plan: in other words, have a backup plan. (c) Don't connect your self-worth to your admissions results—there are more important things in your life than getting into the graduate program you think you love.