I'm addicted to newspapers and other news sources on the web. News abounds on the web but no web-first source for me. In fact, I think the main attraction of the global aspect of the web its that one can easily home in on local-interest sources. The last few years my favourite sites have been the following:
Breaking news is usually brought first by the Dutch teletekst (ceefax equivalent, but in Dutch)
For UK-centered news I read the Guardian and the BBC
For Dutch news I read de Volkskrant and the aforementioned teletekst. Het Parool is not bad either but I read it primarily for news about my home town Amsterdam.
For US news I mainly read the Washington Post and the New York Times. Of course there is CNN as well, but normally I avoid it without knowing exactly why.
For French news I read Libération
I used to read NRC Handelsblad and Le Monde on the web, no bad words about content but I stopped doing so because their sites contain too much javascript and other crap. Le Monde I can easily buy in "handheld" format, but the NRC has lost me as a reader. My current favourite in terms of web design is de Volkskrant: few advertisements, good use of style sheets so that I can even read it using my text-only browser!