Fultonhistory.com but it's mainly for the NY metro area

There's also nyshistoricnewspapers.org, which has archived newspapers from most NY counties.

Ask at the reference desk of your library. If you're affiliated with a university, they're certain to have access to a number of databases, and they'll help you through it. Most local libraries should also have some sort of access, at least to the more notable papers. In most cases, I'd think they'd be able to set you up with internet access, so everything would be downloadable or screen-capturable, although you might have to be physically in the library to log on. You might also check the Library of Congress for relevant materials, if the US is your focus.

If you have a university library near you, check to see if they have access to LexisNexis Academic. Its a database of all major domestic newspapers (over 350 in total, and that's just newspapers), and would be your one stop shop. Most major universities will have access, and generally those libraries are open to the public.

I know that when a particular country or language isn't specified, people usually assume that the questioner is asking about the USA, but I just wanted to make sure that that is what you're interested in?