I use newspaperarchives.com, it's $29.95/month for unlimited article downloads. It's a useful resource but is limited to smaller publications; no NY Times, Los Angeles Times, Chicago Tribune, and Texas is a virtual black hole unless you're looking for info on West Texas.

Augmented with The Library of Congress Newspaper Archives, and SmallTownPapers.com and you've got a decent cross-section of publications. As mentioned before local historical societies are always an excellent resource...