I check both the BBC News website and the website of our local rag (the East Anglian Daily Times) at least a dozen times each day. If there's an interesting story on the BBC site about, say, something happening in Australia, I'll probably try to find an Aussie newspaper website that's covering it too, in the hope of learning a bit more about it. (Similarly, I'll consult the CNN website, or perhaps that of the New York Times, or similar, for more in depth coverage of stories in the USA).
I sometimes buy The Times on Saturdays but that's mainly just for the puzzles section; I'll often leave the newspaper itself unread. However, on the rare occasions that I do like to have a print newspaper to read (on a train, for example), it will usually be The Times that I purchase.