Check to see if the newspaper article you seek is available online. Many publication companies now upload scanned copies of old newspaper articles. At the Google News Archive, researchers can use keywords to search for stories up to 200 years old. Newseum and the Library of Congress also feature online archives.
Consider signing up with a subscription site like Ancestry.com or NewspaperARCHIVE. For a fee, these companies provide access to periodicals as far back as the 1700s.
Stop by the newspaper office, if you have a particular paper in mind and the newsroom is located within easy traveling distance. Ask to view volumes of archived articles, which are usually bound into large books by year. Most offices will allow you to make copies.
Visit public buildings like libraries and county offices. Many counties keep bound archives of newspapers published in the region. Long before the Internet, public and college libraries put local newspaper articles on microfilm. The small photographs are viewable with a microfiche or microfilm reader and you can often copy, scan, print or save them to a flash drive.
Consider traveling to a larger library, such as a federal or state institution, for a wider selection of old newspapers. In addition to an online selection of newspapers from 1860-1922, the Library of Congress is home to a Newspaper Reading Room that contains more than 9,000 newspapers and 70,000 periodicals spanning more than 300 years.