RSS is a tool that makes navigating the internet easier and more efficient.
The simple way to know whether a site has changed (i.e. with new content) is to check it yourself: fire up your browser, go to said site, look for news. This might work without much problem with just one site, but quickly becomes unwieldy with tens or hundreds of them.
With RSS you can select the pages you like and be warned when they change.
It is really easy to get started. First of all, install an RSS extension for your browser (example: RSS for Firefox, RSS for Chrome)
Once you visit a site you deem interesting, look for an icon that looks like this:
now click on the RSS symbol on your toolbar, then on «Find Feeds in Current Tab», subscribe; you are done! From now on you will be notified when the author(s) add something new.
Edit 21/04/15: some sites don’t advertise their RSS feeds with the RSS icon, even though they are present. You can still access them with the above-mentioned method (example: with Firefox go to this Flickr photostream, «Find Feeds in Current Tab» will pick an RSS feed).
RSS to mail services are available but I have never been happy with them.
Let us see what can be read via RSS:
Reuters News RSS Feeds provides an ample selection of news to be chosen from (from “World news” to “Noncyclical consumer goods”). Another popular source of news is BBC News feeds.
Many blogs provide an RSS link. As an example, I follow Point 'n' cook, a culinary blog.
Youtube offers RSS feeds, though it is not an advertised feature; check for feeds in any user page (example).
If that does not work, just format your feed in this fashion:
https://www.youtube.com/feeds/videos.xml?user=GailSibleyAs of 16-Jun-2013 Twitter stopped supporting RSS feeds, but if you really need them you can check out home-made scrapers, or (if you do not care about privacy) third party services.