I don't have widgets but since I am a very minor league weather geek, I have quite a few weather sites bookmarked and I use them on a number of available menu bars.
Try looking at WeatherUnderground (wunderground.com). Since they have a network of local personal weather stations and some of those are APRSWXNET (Automatic Position Reporting System WX Network - a "citizen weather observer program"), you can have current weather conditions and forecasting from very near your location. These are amateur computerized stations for the most part but they provide a lot of information. You can easily save multiple favorite areas too. These are good for current conditions especially.
Now, one really cool thing they are working on is incorporating StrikeStar into some of their radar maps. That is a system of lightning detectors. There are some stations in the system already that let you view lightning in the area overlaid on a NEXRAD radar map. You can also watch loops of the weather radar to see what the weather looks like as it develops. Helps to plan your outings as well as to tell when to gear up for potential power outages. Also, for lightning info, Google "lightning detectors" and your area to see if there are any other sites available.
NWS (National Weather Service) is of course part of NOAA and their weather data collection sites are often located at airports. There are usually a few Forecast Offices in any state that coordinate data and issue forecasts. They're the official government forecasters and you can target small areas for weather. However, they don't have data collected as close to home as you'll find on the WeatherUndergound. NWS current condition reports are a little more general but their forecasting is sometimes a bit better than WeatherUnderground. YMMV.
Just my opinion but I have learned to never trust any weather forecast that is for more than maybe as little as 36 hours away, no matter who puts them out. The weather systems are still too complicated.
There are also websites that will let you track tropical storms, hurricanes, etc. There's a lot of information out there and most of it is easy to understand and to follow.