Genealogy research is easily begun by asking family members if they know any family stories about preceding generations. Did relatives ever mention such things as where they had lived before; were they in the armed forces, or any other such information? Any little tidbit of information can be just the lead needed to give direction to the search.
Search public records at the courthouse for things as wills, birth and death certificates, property deeds, marriage licenses, divorce records and more. Depending on where the search is being done it may be possible to hire someone to help for a nominal fee.
Many public libraries have genealogy sections. There can be found information about the families of the county and state that the library is in. Beyond that there may be genealogy books from other states, books specific to certain surnames, maps, etc.
Genealogy sections are sometimes found at local historical museums.
Some LDS churches also offer genealogy research centers.
Visit cemeteries and look for surnames of the same time period as the name you are researching. Gathering information surrounding other families may help answer questions about your own.
Remember names on records are sometimes misspelled.
Above all else HAVE FUN!Keep looking, there’s more: FamilySearch offers 140 free online genealogy courses