The iPhone's News app allows for individual channels to notify you when there are breaking stories. You can toggle each channel on or off from the Favorites section. To do so, open up the News app and tap on Favorites. Tap on the bell icon in the top left, then you can toggle the channels individually.
Apple News App
In an attempt to compete with other online news sources, Apple replaced its Newsstand with the News app in iOS 9. News pulls stories from RSS feeds and puts them together in a magazine style layout, making for a much more user-friendly experience than its predecessor offered. Although it was a vast improvement, the app hasn't quite gained the traction Apple wants, thus the new round of improvements found in iOS 10.
Even after you set up and customize your Apple News app it still may serve you articles from sources you do not want to read. You can easily mute these sources by tapping an article in your news feed and swiping up. This will open a menu screen with the options Like, Save, Share Story, Mute Channel. Just tap the Mute Channel panel and stories from that source will no longer appear in your For You or topic feeds.