For current newspaper access, your best bet is either to look up the newspaper title in our catalog or search Lexis Nexis or Factiva. For alternative press, you can also try Alt-Press Watch and Ethnic News Watch.
Historical newspapers can be more tricky to access. Many have not been digitized, so you may have to look at microfilm or print copies. Some have not even been indexed which means you have to figure out a date or range of dates and browse manually for information.
Frequently, historians want access to circulation data. This is not easy to get but contemporary circulation data for many papers can be found at the Audit Bureau of Circulation's website: http://abcas3.accessabc.com/ecirc/index.html. The ABC's historical data reaches back as far as 1914 but is only available to Audit Bureau members.
Here are some good historical newspapers available online: