I don't always have time to read newspapers, but if I do I usually buy the Independent... though in reality, I'd read almost anything bar tabloids, and especially not the Daily Mail.
However, I read (many of, not all of) the online articles on BBC news and the Independent pretty much every day, and sometimes also read the Times online and even the Guardian online articles.
I think having only one source for news is a bad thing, but, so long as you're able to stay open minded and recognise all of the choice verbs+phrases designed to sway you a particular way, reading the news from several sources is definitely a good thing.