Most news sites offer what's called an RSS (Really Simple Syndication) feed, which is basically a dynamic list of new articles and headlines. Some RSS feeds contain part or all of the article in question, others just contain a headline and a link to the main article. An RSS feed aggregator lets you access a bunch of these feeds, which are updated as the sites are updated, in a big list. Byline is a good example of a solid RSS feed reader. You can then just click on the articles you want to read to go to them.
If you want to try it out on your computer first, check out Google Reader (google.com/reader)There should be instructions on there to help you figure out how to use RSS, and a lot of iPhone RSS apps will be able to sync to your Google Reader account, so any feeds to subscribe to on your PC will reflect on your phone.
You can also
check here for more info on RSS.