I use an rss reader (am currently using netnewswire but will be changing since their new use of google reader makes it utter garbage) so I don't look at sites per se. BBC doesn't print enough in the rss feed so I see their site more, but I also subscribe to the Guardian and Google News for my mainstream news. A bunch of others for other sorts of news - tech., academic, etc.
I just go to the Google home page, and then click on "News".
You'll have the same story, but it's written by 100's of different news agencies from around the world, you get everything, from tabloids to broadsheets, and it's all free, so you have a choice of how you want to read you daily news.
On a personal level, I would suggest avoiding the tabloids