News Items go on the front page, it is customary to leave old news items and just keep adding new ones, the old ones will help with google rankings, bringing more traffic.
Go to the news item page, click on edit, scroll down to the bottom of the page and click on Delete.
Flyers automatically go in the Flyers category.
Go to the flyer's page, click on edit, scroll down to the bottom of the page and click on Delete.