Where can I find current weather data from radars, satellites, instruments, and forecast model output?

For the United States

See this website for visualizations of weather data for noncommerical use by researchers, aviators, and weather enthusiasts, produced by NCAR's Research Applications Laboratory. Please read the site disclaimer and enjoy exploring the data:


International

The World Meteorological Organization aggregates city forecasts produced around the world by official meteorogical and hydrological services here: