If you live in DFW, you can get an instant weather report at Local Time and Weather, or try NBC's Intellicast Weather for DFW. If you're not in DFW, you'll probably be able to find your city starting at NBC's Intellicast Home Page.
If you're really into news, go straight to News, Weather, Radio where you'll find links to CNN, USA Today, Associated Press, Reuters, Dow Jones, US News, Time Warner, BBC, Internet Radio... and much, much more. Also mentioned above is Radio. You can listen to recordings on the web. This is cool. Download and install the RealAudio Player, go under preferences in Netscape and set the MIME type audio/x-pn-realaudio to the RealAudio Player. If you're using System 7.1 or earlier, you'll also need to install Sound Manager 3.1. Then, go back to "News, Weather, Radio", and click the "AudioNet" link. While I'm writing this page, I'm listening to Barry Switzer and Larry Brown's press conference straight off the web. A web search for RealAudio will turn up lots of other things to listen to!