What sort of article are you hoping to publish?
Newspapers rarely print unsolicited articles from freelancers. Other than the editorial pages, they normally print news, i.e. accounts of current events based on the reporters eye-witness experience or interviews with eye-witnesses. Other than the editorial page, newspapers don't normally print essays, and I've never seen a newspaper print a short story.
There are freelance reporters who travel around looking for news and writing up stories. If that's what you're up to, you could try to submit those kind of stories. You could try to get picked up as a columnist for the editorial page.
As Rich Moy and Tave point out, if you're just starting out as a writer, the New York Times and the Washington Post are probably not your best bets. These are very big publications that lots of people want to write for. I don't mean this to sound snide or mean, but it's like asking, "I just graduated high school. How do I go about applying for a job as president of General Motors." Yes, if you're very good and/or have the right connections and/or are very lucky, you might manage it. But it makes way more sense to start with more humble objectives and work your way up. Start with a small local publication.
Magazines are much more likely to accept freelance submissions. The trick here is to read the magazine and find out what sort of articles they publish, then write something that fits. Most magazines accept freelance articles. Though some don't: you'd be wise to check out the magazine and their web site if they have one to see if they give a clue on this. I got my start in writing by publishing in computer magazines.