The perennial Google Reader mention makes me feel obligated to mention that there are still many blogs alive and well and with RSS feeds. You can still do it, and there are so many reader applications. My recommendation is The Old Reader (basically a GR clone), and Reeder on iOS (syncs to stuff like The Old Reader)

Yes there are no RSS feeds for Twitter or FB. But there’s a lot for a lot of places. I’ve never found a website where I wanted an RSS feed and where it wasn’t available (absent maybe FB, where a feed for my closest friends would be nice)