When you've decided where you want to search, then you'll need to find out what newspapers were being published in that area during the proper time period and where these papers, or copies of them, might be found. If necessary, find a map for the proper time period and make a note of the nearby counties and the names of as many nearby towns as possible. Many times one of those names will be part of the newspaper title.
As is most often the case, start with a library. Armed with the name of the county your ancestor lived in, and the names of nearby counties, look for published histories on those counties. These histories almost always name the areas first newspaper or newspapers and often the history of that paper. It is important to check those surrounding counties to narrow down the newspapers that actually covered the area during the time your ancestor lived there. With initially sparse populations, the nearest town with a newspaper may have been 20, 30, 50 or more miles away.
If you are lucky enough to have a cheap telephone plan, you may be able to obtain the name of the proper newspaper(s) with one quick call. City and county libraries often have that information on hand and, sometimes, even have access to microfilm copies of the newspaper in question.
Bear in mind that many newspapers changed ownership and often the newspaper's name, over the years. If the newspaper is still being published, you should be able to find the correct name in the "Gale Directory of Publications and Broadcast Media." In one form or another, this directory and its predecessor have been published since 1869. If the paper ceased publication, its name can normally be found in prior volumes.
A specific source for tracking down newspapers, and where those newspaper files are located, are union lists. Union lists are catalogs normally arranged by state and communities. They provide information on the newspaper offices themselves, specific libraries, historical societies, and even private collections that have the files. Again, you will want to track these lists down through a local library.
Union lists include the following:
"History and Bibliography of American Newspapers, 1690-1820," in two volumes, Clarence Brigham, Worcester, Massachusetts: American Antiquarian Society, 1947.
"Newspapers in Microform: United States, 1848-1983, in two volumes, U.S. Library of Congress, Washington, D.C., 1984.
"American Newspapers, 1821-1936: A Union List of Files Available in the United States and Canada;" Winifred Gregory, 1937; Reprint. New York: Kraus, 1967.
If you cannot find these resources in your local library, try the nearest university library and ask for their reference department. In addition to providing names of the newspapers, and the time period they were published, they should also tell you the frequency of publication. This may help you date the time of an event or even provide the date of an issue you need to read.
Naturally, most of you will do some research on the Internet where you will find copies and abstracts of newspapers that others have thoughtfully posted to share. But, don't limit yourself to the World Wide Web. Track down those historical societies and genealogical departments in libraries. Call them, or send a letter. Many hidden treasures are tucked away on a shelf and you'll never know they are there if you don't ask.