I am going into counseling psychology so I did ask the basic questions about research, qualifying exams, and internships. I dont know what you are going into, but those should be the basic questions (i.e. what do the next X number of years look like for me?). Also I asked about the city, what there was to do, what does student fees cover, where are good places to eat/study/drink coffee/exercise on campus, how are classes arranged (block schedule? everyday? this school that i visited didnt have friday classes, so that was interesting), what TA/RA-ships are available, am I able to teach courses, how are classes taught (for this school they focused more on papers and presentations instead of exams), what kind of resources does their library have, etc., etc., etc.
Hope this helps a little!