NEXRAD data can be accessed using the NEXRAD Inventory Search tool, or alternatively, from the NCDC HDSS Access System (HAS) website. The Level II data are the base digital data (mean radial velocity, reflectivity, and spectrum width) at the full spatial and temporal resolution of the radar. The Level III data are the radar product data generated from the Level II base data. For most educational purposes, the Level III data should suffice.
Satellite data can be obtained through the Space Science and Engineering Data Center at the University of Wisconsin-Madison.
The Department of Atmospheric Science at the University of Wyoming has a fairly comprehensive and easy-to-use sounding archive.
Surface observations and other weather data can be obtained from the NCDC.
Here are some resources that are freely available at academic and government institutions:
The Integrated Data Viewer (IDV) from Unidata is a Java(TM)-based software application that allows analysis and visualization of meteorological data. It can be used to display radar and satellite data, as well as gridded model output, soundings, and surface observations. It accepts a variety of data formats, and does not require data pre-processing. Thus, I find IDV particularly valuable for radar-data display.
The NCDC Weather and Climate Toolkit is Java(TM)-based software that can also be used to display (Level II and Level III) radar data.
The NCAR Command Language (NCL) is versatile, multi-purpose software. It is useful for general data analysis and display. Example NCL scripts are provided with several of the exercises on this website.
The CM1 (Cloud Model 1) is a 3D, non-hydrostatic numerical model, designed primarily for idealized simulations. It was developed and maintained by Dr. George Bryan at the National Center for Atmospheric Research (NCAR). The installation and implementation of CM1 is straightforward. Suggested experiments with CM1 are given as part of several of the exercises on this website.
The Weather Research and Forecasting (WRF) model is a 3D, non-hydrostatic mesoscale numerical model.