Your favorite news sites and blogs probably have feeds that you can subscribe to. The easiest way to add them is to search for them in Google Reader -- just click Add subscription and enter your query. If you can't find the feed that you're looking for, here are some other ways to find and subscribe to feeds:
Let Google Reader Find It
Copy the address of the website (for example, http://googlereader.blogspot.com) and paste it in Google Reader's Add subscription box. If there's a feed associated with the page, it will be added to your subscriptions.
Modern browsers like Firefox will usually display an icon 
in the address bar if the site has a feed.
Search for subscription links
If the orange icon is nowhere to be seen, try looking for links with the following terms on the page (in Firefox, you can choose Edit->Find in This Page...):
- subscribe
- syndicate
- feed
- rss
- xml
- atom
Many of these are technical terms for feeds you can subscribe to. You can also look for images or buttons with these terms on them, for example:
Click the most obvious link or icon. Copy the address of the page you're taken to, and paste the link into Google Reader's Add subscription box. In some cases, the page may look unreadable, but it will look normal in Google Reader.
Some sites have multiple feeds, so make sure you choose the main feed and not a comment feed, for example. If the site gives you the choice between RSS and Atom, choose either one; Google Reader understands both.
If All Else Fails
It's also possible that this website doesn't have a feed yet. Try finding a contact page and email the site owner to suggest that they add a feed.