Look around the room. Do you see an elderly relative? If so, cough politely and say, "Sir or madam? I have a few questions about the family." You're looking for two kinds of information: places and names. What was the name of the town where they grew up? What were their parents' first names? How many siblings were there? What was their mother's maiden name? Is there a family gold hoard? Information like this disappears surprisingly quickly from one generation to the next, so pay close attention to what the relative says, as this is literally the biggest chunk of information you're going to get all at once.
Once you've exhausted your new favorite relative and she's snoozing peacefully in her favorite chair, it'll be time to hit the Internet. If you're researching people in the United States, the National Archives has tons of information about where to start looking, how to read census records, and even how to preserve all of the newspaper clippings and stern-looking ancestral portraits that you're going to find.
The National Archives also has census, immigration, land transaction and military records. However, most of what's available online is almost exclusively in the form of scanned microfiche documents. Because the entries are handwritten and indexed by place rather than name, it can be a daunting task to squeeze out any information unless you know exactly what you're looking for. To uncover the details, you may have to visit them in person – the National Archives is open to the public, and there are branches all over the U.S.
Ancestry.com and HeritageQuest.com have digitized versions of census records that you can search by name. Though these sites are subscription-based, they offer free access from any National Archives facility.
FamilySearch is a project maintained by the Church of Jesus Christ of Latter-Day Saints that offers free access to digital records from all over the world, and you can search by name. If you register with them, you can also maintain charts and photo albums online as your research progresses.