Are the Newspapers Online for Free?
Northern New York Historical Newspapers
These newspapers are for several northern New York Counties and I have had some success searching there for my wife's French-Canadian ancestors who came to Clinton County, New York, in the 1850s. I have found several references to a great-great-grandfather who moved to Chicago for a short time, including a death notice for him.
Quincy, Illinois, Newspapers
There are several Quincy, Illinois, newspapers on this site. I have located several previously unknown tidbits, including a child born in the early 1870s whose father was not who we thought it was and a sibling of a great-grandparent who apparently had an alcohol problem.
Both sites are free to use and both have full text searches. The Library of Congress has a small smattering of newspapers online for free on their site as well.
Ancestry.com and Genealogy Bank have some nice newspaper collections on their site for members, but there may be some newspapers online at no charge. Keep in mind that free sites, while you should look for them, are more the exception than the rule. There may be a site for your areas of interest as well.