Begin with what you know
Genealogists use a few standard forms to help organize their data, the Pedigree form and the Family Group form. These can be printed up using genealogy software, or picked up at a library or genealogy library. Fill out a pedigree chart on yourself. This will include the names of yourself, your parents, their parents, and their parents. You'll also want the dates and places of their births, marriages, and deaths. For each couple on the pedigree, fill out a family group form to include all their children. If you have genealogy software, you can enter in all this information instead of writing it down. Just by looking at the filled out (or printed out) forms, you can tell what information you are missing.
Decide what you want to find out
Chances are, your pedigree chart has a few holes in it. Decide which bit of information you want to discover first. Perhaps your pedigree chart is missing the names of your mother's mother's parents, and you want to know who they were. First, are there any living relatives that might know about Grandma's parents? Is your mother or grandmother still around? How about aunts or uncles? Are there any family records, such as a family bible or old newspaper clippings, in the care of a cousin? Just asking around can answer questions that save hours of research. When you've picked the brains of everyone who might know something, it is time to start searching public records.
Determine what sources are available to help you, and search them
Vital records, such as birth, marriage, and death certificates, are the most reliable sources because the event was recorded when it happened. Some websites have searchable records, for example, Vitalsearch-ca.com has a searchable database of 1905-1995 California birth records. Some private detective websites, such as Intellius.com, have vital record searches available for a fee. Older vital records may be searchable online at specific genealogy sites, such as Ancestry.com (fee). If you can't find what you need online, most of the legally available pre-1909 vital records have been microfilmed and can be ordered through a Family History Center. You also may be able to order an official copy of a close relative's birth, marriage, or death certificate from the state or county where it occurred. If you do that, request an exact copy of the original certificate to get all the juicy details. Deaths reported to Social Security are public record, and there are several genealogy websites with a searchable version of the Social Security Death Index (SSDI)
You can also learn a great deal from an obituary. Often, they include information about at least 3 generations of a family. Many newspapers now have the past few years of obituaries searchable in the archive area of their website. For older obituaries, you can often order microfilms of old newspapers through Inter-Library Loan (ILL), or try Ancestry.com (fee) to search and view the newspapers that they have indexed and scanned. Cemetery records are a great resource too. Check with a genealogical society (most have an e-mail address or a website) in the area where your ancestor lived to ask about what cemetery records are available. Many societies publish these records as books and sell them. Others may be searchable online.
Another great source is the U.S. Census, which has been taken every 10 years since 1790. In 1850, U.S. Census records began to record all members of the household, instead of just the head of household. Slightly different information was collected for each census. Some census years are now indexed and searchable online at Ancestry.com (fee) and Genealogy.com (fee). The 1880 U.S Census is searchable for free at Familysearch.org. Unfortunately, most of the 1890 U.S. Census was lost in a fire, so only bits and pieces of it are available. U.S. Census records can be manually searched on microfilm at National Archives, Family History Centers, large genealogy libraries or online at Ancestry.com (fee). U.S. Census records are kept private for 72 years, so the most recent one with public access is the 1930 census. Other countries took censuses too. Check the Family History Library Catalog at Familysearch.org to see if there is one available that could help you. The 1881 British Isles, and 1881 Canadian census are searchable on Familysearch.org.
If your ancestors haven't been in the U.S. for very long, tracing the family in their homeland may not be as intimidating as you might think. The Ellis Island website (ellisisland.org) has indexed their immigration records for searching online. If your ancestor came from Europe, and you know their hometown, tracing their parentage can be as easy as searching a microfilm of the local parish register, and learning a few words in the native language. Many European countries have meticulous records that were kept by the church. These can date back into the 16th Century. European and other non-U.S. microfilms can be searched at the Family History Library in Salt Lake City Utah, or loaned out to Family History Centers worldwide. Find out what is available online at Familysearch.org by searching the Family History Library Catalog. Another great website it WorldGenWeb.org. Information specific to certain localities may be available through its links to member sites. The World Gen Web project is done completely by dedicated volunteers, and is free.
Some genealogists submit their family data to websites where it can be searched online, such as Ancestor File or Pedigree Research file at Familysearch.org. Another ambitious project is OneGreatFamily.com (fee), where these individual databases are connected to each other. You may find that an unknown cousin has posted some of your family tree online. Always consider other research to be undocumented until you have confirmed their data with your own sources. There are countless errors in many of these online family trees, but they can be a great starting point for further research. For your own sanity, always document where you got a piece of information so you can go back and check it again if you find conflicting information from another source. As you encounter conflicting information, consider the source. A copy of a birth certificate or census is a much more reliable source than undocumented data from a website.
At USGenWeb.org, you can find links to genealogy websites related to the town or county that your ancestor was from. These sites usually list sources that are specific to that area, and how to access them. Most of these locality-related websites take "Queries" on their message boards, where you can post specific questions about a family name or a specific ancestor from that area.
Some genealogy websites are set up by good-hearted family history hobbyists, others are set up by businesses or organizations. Small sites can be useful for finding obscure information, and are usually free, but are often hard to search. Some may require a free registration. There are websites, such as Cyndi's List (cyndislist.com) that contain millions of links to specific genealogy sites both small and large, that may help you. There are a few commercial sites that have massive amounts of information that are very easy to search, but they usually charge a fee. If you want to do most of your research from your home, you'll want to consider a premium membership to one or more of these commercial sites.
There are many other records that can be searched. Land records or probate court records (wills) can be valuable in learning about the descendants or surviving spouse of a person. Local history books often have mini-biographies and of local citizens who paid to get their name in the book. City directories are basically old phone books, and can tell you where the old family home was or a bit about a family business. Newspaper articles other than obituaries can give some color to the dry facts about a family. As you become more familiar with the area your ancestor was from, even more creative sources will emerge.
Learning about the people in your family tree may become more fascinating as you go along. Some say that genealogy is "addictive." As you search, you may discover a link to a famous person, a Civil or Revolutionary war veteran, or better yet an interesting "black sheep" in your family. Whatever you find, you can be sure to be the resident expert at that next family reunion!