The original invitation message, sent out in December 1999 to recruit members for the Spatial Systems Laboratory, gives a partial overview of what we're doing.
Professor Propp created some tutorial documents on particular mathematical topics related to chip-firing: the geometry of multi-dimensional lattices, the definition and use of toppling matrices, and the use of determinants to count spanning trees of rectangular grid-graphs. (These files are not intended to serve as a systematic introduction to chip-firing, but they give useful supplemental information.)
Some other short documents (not yet organized by topic) written by the undergraduate members of SSL are available from a directory currently being maintained by Mark Chapman.
I have also assembled a bare-bones bibliography that should at least help you get started in your own explorations.