Whenever really bad weather threatens, StormWatch will provide you with the latest watches and warnings on a national map; you can zoom in for greater detail. It is a site worth visiting when any type of severe weather threatens.
The Storm Track megasite is a combination of two entities: Storm Track Online, the official website of Tim Marshall's 25-year magazine, and Storm Chaser Homepage, the massive chasing resource created by Gilbert Sebenste in 1995..
For tornado information, the National Severe Storms Laboratory and the Storm Prediction Center in Norman, Oklahoma may also prove to be useful sources for further information. And I cannot recommend strongly enough the excellent videos and books available from The Tornado Project; their Web site will update you on their activities.
And if your interest is hurricanes, check out the National Hurricane Center in Miami for the 'official' information. Another excellent source is the University of Michigan's Weather Underground Tropical Page with links to a variety of products and data. To help you track the storms, there's HURRTRAK Professional - a superb commercial professional-level tracking program (~$360 and up). For home users, each of the following shareware Windows-based programs offer their own distinctive approaches and features: Merlin (~$39), Tracking the Eye (~$25), and Hurricane Watch (~$20).