For the record, I'm visiting the University of Chicago as an accepted PhD student (in the geophysical sciences) next week.
In particular, I'd especially appreciate creative questions that few other visiting students ask.
Talk to current graduate students in your lab (even better, working with your potential adviser!)
- Ask about their opinion about how the working environment is - what is the working style of your adviser, what is the average time take for a PhD student to graduate from the lab, and where do most of them land up (as eternal postdocs, or tenure-track faculty positions)
What is the funding scenario - do most people have to be TA's for an extended period/all throughout, or whether RA grants are available? What about conferences - do students get funding for traveling to important venues?
What kind of a social life exists in and around the campus, and what is the cost of living - a Ph.D is a long commitment, and you should remember these "soft" criteria as well, so that you are at least prepared mentally when you enter grad school
I'll expand on my answer as and when other points come to my mind!