Gather your family records and stories. These are your starting points.
Try local or county libraries. Many have a local history or genealogy section. Some have newspaper indexes and other research aids which are only available there. Also, they may have a subscription to Heritage Quest or Ancestry.com with access to indexed census and other records.
Keep track of your research using Ancestor Tree. This will organize your results and provides many ways of viewing them. Use the "Research Assistant" utility to help you plan and organize your research. Your research and plans are then available online as you travel. Use Ancestor Tree's charts and reports to direct your research to "Branches and Leaves", or in other words missing ancestors and descendants of your ancestors.
Try online sources: see our list to get started.
Search databases and the web using variations of the surname you are looking for. Genealogy is inherently historical, so searching for a locality and time period and with words such as "genealogy", "birth records", "death records", "family history", or a name or surname is often helpful.
There are public domain books available at archive.org, heritagequestonline.com (if available through your library), and books.google.com/advanced_book_search.
Local or county historical societies may have genealogy sections. Sometimes there are genealogical societies specifically for this purpose. State and regional historical societies are sometimes helpful. National archives and societies or archives based on country of origin, immigration, service, religion, or ethnicity may also aid research.
Civil and religious records are the standard. Every jurisdiction has its own procedures. Some are freely available on the web. Searchable records on the web are a very powerful tool. Please support organizations that maintain these sites.
Relatives in other branches of your family may have records of your common ancestors. Ancestor Tree gives you a place they can find you and you can collaborate with them.