Qgis or Quantum GIS is among the most widely used Open Source Geographic Information Systems. The current version is Qgis 1.7, which was released on 29 September 2011. Below are a few pointers to get started.
For Mac (Lion and Snow Leopard) use the installation from here: http://www.kyngchaos.com/software/qgis
There are two extra components that are required to run Qgis on a Mac. Those are the GDAL Complete 1.8 framework package as well as the GSL framework, also available on the website above, and which should be installed beforehand. Installation is performed as always.
For Windows, there exists a standalone installer here, which is sufficient for the novice.
More information on different installations and other platforms can be found on the Qgis download page.
There are many Qgis tutorials available, but only a few are using the most recent version of Qgis. Here are two resources I found particularly well done.
This is a manual for a day-long workshop provided through Newman Library at Baruch, CUNY. It covers Qgis 1.5 and 1.6, but here is a link that describes the few relevant changes that were introduced with Qgis 1.7. It is rather comprehensive, as it also introduces basic GIS concepts, projections, and simple spatial analysis and thematic mapping, but provides a very good overview of how one might get started with both, the software and spatial analysis.
Another great introduction to Qgis comes from Berkeley's Knight Digital Media Center. It is focused on how to quickly visualize data in a map. It covers specifics of GIS files and editing, how to join them with external tabular data and also shows the way to make simple, interactive data maps that work on every platform. This tutorial covers Qgis version 1.6, but also addresses changes made in 1.7.
It is also worth searching on Youtube, there are a series of screencasts, that step you through the installation and the components of the software or provide task specific instructions, for example:
Finally, the Qgis Users list: http://lists.osgeo.org/mailman/listinfo/qgis-user