Some people are born interviewers, they just seem to be excellent at asking just the right questions and remembering all the answers. This article is likely not for those people, however I suspect even they may find some ideas here that would improve their abilities to find out nuggets of family history interest from their relatives.
First off, is take something to record your interview with you. Don't expect to take notes. It's hard to keep up and it takes you out of the conversation. Something that records direct to digital audio files is a most wonderful choice. I have a Zoom H4 digital recorder that I use for many purposes and this would be an excellent choice, but there are many choices out there. Just be sure that whatever you use you are able to convert the files to something useful to share with others at some point. Be sure to ask your relative if they mind if you record the conversation. Explain that you want to ask about the family history and about what they remember growing up and explain that you would like to record it so you can get all the details straight and not have to sit and write the whole visit.
Okay, so what do you ask about? So many genealogies are simply dry facts. Name, date, born, died, where can be such a dry distillation of our lives. Try to dig a bit deeper than that. Ask if they could describe what their parents looked like, favorite sayings of a family member, favorite songs. Did they play a musical instrument?: What did they do for a living? What was it like growing up in that family? When did they first get a radio/tv/car/etc. what was that like? Did they ever travel, where, how did they get there, what was that like? Where did they live, why? This gives you some ideas. Try to let the conversation flow.
Now, I just said try to let the conversation flow, but I think you should go in with a list of questions that you would like to hear answers to. Think of yourself as a reporter digging for a story. If new questions come up in the discussion, follow them and see where it leads, but try to get through all your questions. If there seem to be too many questions to get to, you might consider finishing up and not making the visit feel like an extended inquisition. It's common for an eager researcher to be excited about finding out family details and lose track of time. Keep your family members time in mind too though, they will likely be glad to chat with you, but don't overstay your welcome. It might be better to say "I can't believe how the time has gotten away, we've been talking for an hour, I should probably go, do you think we could do this again sometime?" than to have to be told, "Wow, we've been talking three hours, I really need to get started with dinner."
Think about common interview techniques, in other words learn from good interviewers on tv or the radio. Notice how they try to work into open questions to let the interviewee answer with more than a yes or no? Also, don't expect all the answers. Some people don't remember much of what they were told as children about their extended family. You could probably find out a lot about what life was like as they were growing up, about things they did as a family from those that don't remember many family details. That kind of information can be as valuable.
Another good idea is to take what you learn in your initial interviews and do some research from there in the library, courthouse, or elsewhere and fill in some details. Then, go back and ask questions about some of the things you've found. "Do you remember anything about your parents having a child that died young?" Using research to go back and have further discussions can help refresh memories. I remember in one of our family histories it took finding a great-great-great grandfathers name written for some of the great-aunt's and uncles to say "oh yes, that's right daddy had a grandfather named _______." Sometimes you'll be able to use those moments to either simply confirm what you've researched, or even expand a bit.
To finalize, I suggest any family history researcher or genealogist to make talking to their older relatives the first priority in their ancestry research. We often overlook the recent history in search for the older histories, but sometimes it's so much easier to put details into the more recent history if we just take the effort to do so. Additionally, if you spend too much of your time in the library early on, you may regret missing the opportunity to learn more from your older relatives when you had the chance. Enjoy your interviews and good luck with your research.