English prof here, interviewed at MLA last year:
Agree with poster above. Be ready to discuss their campus and students in a way that doesn’t sound like “I memorized this from your website on the plane ride here.”
Research, as much as possible, the fields and publications of the search committee and the department as a whole. Though the mantra is that schools look for cutting-edge scholars, I think the truth can sometimes be closer to they look for candidates who most resemble themselves BUT aren’t perceived as an intellectual threat or snotbag (this is a big statement, does not apply to all schools, Ivies are probably different: Your mileage may vary.)
Be sure you can answer “why do you want to come here” confidently. I’m sure you know this, but don’t ever say “because I need a freakin’ job, dumbbell!”
Gauge if the department leans more toward teaching or research and skew your replies to that end. If it’s a teaching school, make sure you’re clear about your enthusiasm for teaching both intro and advanced courses. Many newbies (like me) end up teaching intros for a while – express enthusiasm and competency! Be sure you can expand on “how would you teach x for first-year students? For advanced grad students?” If you’re in a field with esoteric potentials, be ready for the crazy search committee member to ask how you would teach 12th century mnemonic plainsong hymns via their new online Blackboard delivery system.
For research questions, be clear and direct. Don’t try the I’m-a-grad-student-out-to-impress-and-cite-every-person-in-the-field-I-can-remember. Express enthusiasm and connection to your dissertation, speak in clear language, and outline a few plans for the future ("I really feel that chapter three, in particular, will make for an exciting book project because . . . ") Make sure the committee knows when you will absolutely, positively be done and have the diss defended.
So far as questions to ask: How many undergrads? Grads? Retention rates? Support centers for student learning? Interdisciplinary efforts? I asked some questions about how campuses recruited from diff populations to support their mission statement of diversity and access and what the first-year typically looks like for a new hire, etc. Do not ask anything about salary or benefits. I have some friends who asked when the committee would be making their decision, I wasn’t comfortable asking.
“The Academic Job Search Handbook” was an excellent resource for me. Since I’m anxious about being able to remember stuff, I also made bullet-point 3x5 cards and put aside some time each day to “study” myself.