By historic day, I mean, for example, the before and after of World War II, the reaction of the entry of the US in WW2, or the Vietnam War. I've read some of the Pittsburgh Press, and it's amazing how they viewed and reported these events.
If you are just looking to read them online the two main digital archives I use are:
[Papers Past (New Zealand)] (http://paperspast.natlib.govt.nz/cgi-bin/paperspast)
[Chronicling America (U.S.)] (http://chroniclingamerica.loc.gov)
I believe Chronicling America only goes up to 1922, so WW2 would be not be available I am afraid. However, WW1 or the Spanish American War, or a number of other equally interesting and random things are available. Frankly it can be an amazing resource to just get lost in.
Papers Past goes up to 1945; so that would at least give you a New Zealand perspective on things if you are interested in the Second World War.
Yes, most major newspapers keep archives, sometimes stretching back as far as the 1700s. Some of these archives have been digitised and made available online, but you have to pay for them.
Examples:
I remember that the Le Monde archives used to be free, but what little French I used to have has abandoned me so I can't figure out their website anymore.
Alternatively, you could try your local library, which would probably have archives for local papers and possibly a subscription to an online archive service.
If you are in the US and near a decent university, just visit their library. Most will have database subscriptions to at least a basic set of papers, typically the Times of London, New York Times, Washington Post, and then a few regional papers (Chicago, LA, Detroit, etc.). The holdings in these databases vary but many go back into the 19th century and often cover entire runs. While you won't be able to access them at all from off campus, they are quite often open-access from networked computers in the library.
At my university we not only have access to a half-dozen or so US papers, but printing is free and unlimited.
They are available, but a lot of them are behind paywalls, or require you to be associated with a university that has access to the database indexing them.
This one is for British newspapers, and there is a free trial available.
Proquest runs one of the better ones, but again it requires a fee. There is an option to request a trial but I don't know how long that would take.
As a plug for NZ history, PapersPast is a free access database, with newspapers from 1839-1945 from around New Zealand, and is regularly being added to.
Wikipedia also has a handy list of freely available US newspaper archives. They are scattered. It appears the Library of Congress also runs a program called Chronicling America which has a huge number of newspapers available.
Lastly, I'd suggest you check your local libraries(public and university). There might be a way you're able to look at their microfilm collections and see more recent era newspapers.
I'm not sure if the scope is exactly what you're looking for, but there's an amazing resource over at fultonhistory.com that you might check out. There are over 28 million newspaper articles from those areas bordering Lake Ontario on this site. It gives good insight into the non-major-city lives of those people from more than a century back over national issues in a part of the country that, depending on the time period, was on the verge of being very important or very obscure. The websites UI leaves a bit to be desired, but it's manageable.
The information on the area around the Oswego River alone is invaluable due to Oswego and Fulton's varying importance back in the day.
Many of the newspapers in museum collections have been digitized and are on searchable archives. The problem is that many of these archives aren't accessible except through university or library systems. Some are free, like Chronicling America that covers 1836-1922. If you search online for "newspaper archives" many libraries will have link to the more local newspapers that have free databases.
http://fultonhistory.com/ has a bunch. I found my grandparents' engagement and wedding announcements from the '50s.
google.news has a date parameter in the advanced search funtion. you can use this to easily dig up old news articles!
If you live in a major city with an old newspaper, check the public library. I live in Boston and the BPL has newspaper microfilm going back to the late 1800s. It's wild to see how papers evolve over time. And it's all free!
The Internet Archive has a pretty large collection of newspapers. Their search functions aren't the greatest, but there's a lot of neat stuff to be found.
No one has mentioned Trove yet, which is for Australian papers, seems to be free (I used it to research several civil wars in the Pacific over the past two months), and goes all the way back to 1803.
I'm certainly no historian but I've had an interest in this topic for a while for genealogical and other reasons. At some point a few years ago Google scanned the archives of several hundred newspapers. Many of them are from smaller towns and others are incomplete but nevertheless its a pretty impressive archive in my opinion. The link is below. Hopefully a paper you like is included. Enjoy.
EDIT: Meant 28 Million pages; mobile won't let me edit
The newspaper collection at the American Antiquarian Society is pretty amazing (five miles of shelf space and all pre-1876). Unfortunately you cannot view the collection online without a subscription through somewhere like a university, but you can absolutely visit and use the real thing for research. Info on the collection: http://www.americanantiquarian.org/newspapers.htm
Proquest Historic Newspapers has a lot, and is usually available through your local library system or a nearby metro library system.
Most newspapers themselves have libraries/a librarian who organizes and preserves old paper copies. I've had good luck going to the newspaper's office and asking to see their old, hard copies. Granted, this is a bit out of the ordinary, but I knew the editor and this was a small town weekly. Major papers as well will have these archives on site, too.
https://news.google.com/newspapers?hl=en
Google has hundreds, if not thousands, of newspaper archives.
I currently work microfilm at a Library, and this is basically my job description!
Google News Archive has a massive collection of news articles from a wealth of newspapers going back into the 1770s (and perhaps even further, though I haven't stumbled across anything from before 1779 yet).
Used to be you could search historic articles by going to Google News, then clicking on "Search tools", "Recent" and "Custom range" and then specifying the exact dates and typing the words you want included in the search but it's been temporarily disabled for more than a year while they update it. Back in December they promised the updated version would be out soon.