Hi Gene -
I don't have answers to most questions in here. Finding a good free GIS is something I've been trying to figure out too. QGIS (mentioned above) seems fine for basic stuff, although I've been having trouble figuring out how to get beyond the basic with it.
Just a clarification for you (in terms of the techy terminology). Raster is one of two main types of spatial data. The other type is Vector. Raster files are a continuous surface of pixels (think of satellite images) while Vector files are collections of points, lines (roads, rivers...), and polygons (areas like states, countries, your backyard, whatever). It sounds like what you want to work with is Vector data, so just ignore whether or not programs can deal with Raster. I think all programs can do Vector, Raster is the one that can only be done by some. So basically any program out there should be able to do what you're looking to do (infant mortality by country or something like it).