The ESRI Training Center offers a number of low-cost, on-line, self-paced courses and online certification programs. Sample software is occasionally available. They also host the introductory website, What Is GIS?
CSISS (at UC-Santa Barbara) offers SPACE workshops on GIS and spatial analysis, plus links to GIS literature.
A Gentle Introduction to GIS uses the free and open-source Quantum GIS software, but the concepts are valid for any GIS.
The Harvard University Graduate School of Design has a detailed Online GIS Manual covering many processes and concepts.
Visit the GIS Lounge of the American Association of Geographers for software, data, and other types of support.
The GIS Cafe-Universities is another web community/clearinghouse for all kinds of GIS information and activity.
GISGeography posts articles on how to accomplish various GIS tasks.
Penn State Department of Geography has an online course, Nature of Geographic Information.
There is also an online course on Spatial Analysis by the authors of the well-known text on geospatial analysis.
The Northeast Arc Users Group sponsors a yearly conference for all users of ESRI software in the Northeast.
The NYS GIS Association sponsors an annual conference and provides updates about GIS activities across the state.