A related option is iGoogle, which many people use as a Start Page when their web browser launches.
iGoogle allows you to add RSS feeds as well as thousands of "gadgets." To add an RSS feed,
Use the Main Page for seeing the latest posts out of all you subscribe to.
Use folders to gather feeds on similar subjects, then click a folder or a feed to see just those new items.
For more details, visit Intro and Tips for Google Reader
To subscribe to a feed, click on the red SUBSCRIBE button in the top left corner of Google Reader and copy/paste the URL of the feed you want.
The older posts don't totally disappear once you've read them. You can view the contents of older posts by clicking on the name of the feed in the left sidebar, then scrolling down.