How big is the department, and how flexible is it?
Don’t limit your assessment of a department to the particular people working in your current field of interest. Your interests could change, so it’s better to be in a department that is (a) large enough to accommodate a wide range of potential interests, and (b) even more important, is flexible about the kinds of topics and projects that students can pursue. Some departments tend to be more conservative about research methodologies, dissertation structure and thesis topic. I was very lucky to put together a committee that let me work on an unusual topic, and with a non-standard structure.
What kind of funding is available for graduate students?
Make sure to get a read on the amount and stability of funding available in each department. This information is crucial if you get admitted, but want to negotiate for an improved funding offer: there’s no point asking for more than $12k if that’s the department’s highest level of funding. Also note that some places give good offers for year 1 but don’t guarantee continued support.
How long do students in this department take to complete the degree? How long do students in my planned subfield take?
That’s not-unrelated to the question of funding levels. Some departments end up providing so little money that people teach tons and take forever to complete the degree. Also find out when people take their general exams/field exams (different departments call them differen things), what the exams involve, and how long people take to study. E.g. at Harvard (when I was there anyhow) generals were oral-only, at the end of year 2, and people only take a semester to study. Columbia has orals and writtens and when I was applying, people take all of year 3 to study. So that adds a year to the degree right there.
My complete guide to grad school applications:
Who I am: my faux qualifications for dispensing grad school advice
My story: how I survived the application process
My results: where I got in, and how I got funding
Questions to ask yourself: things to think about when applying
Questions to ask departments: things you need to find out for your applications
Acing the application: my tips for winning at the application game