Many GIS students are familiar with expensive, proprietary GIS software packages. Mainly, this is because these are the software packages they were taught in school, or learned in their first job. However, when leaving the friendly confines of school, the true financial cost of these programs becomes evident. In many cases, this cost is too high to justify their purchase for use in a a small business or for home 'recreational' use, and leaves GIS users looking for alternatives.
Fortunately, there are many fantastic options, and luckily some of them are even free!
Enter the world of Open Source GIS software! Open Source software is community maintained, freely downloadable, and generally usable for most purposes (it pays to check the licence agreements carefully before using them for commercial uses, but for the most part you should be ok.)
So, where should you begin your study in the world of Open Source GIS?
For basic 'desktop' GIS data display, query and editing, Quantum GIS (aka QGIS) is your best bet. It's full-featured, user friendly, and development cycles are quick. For more advanced analysis, you'll want to check out Geographic Resources Analysis Support System (GRASS) which can also be used as a plugin inside QGIS, making for a very powerful analysis and modelling package.
For behind the scenes data conversion/translation/etc. you will want to take a look at GDAL (for raster data) and OGR (for vector data). These packages are most easily used/installed via FWTools, and run from the command line.
'How about data formats' you may ask? Most open source GIS packages read and write all common GIS formats, so it shouldn't really be a problem. However, if you are also looking for open source formats, start with PostGIS, which spatially enables the Postgres database, or SpatiaLite for a more portable file-based format.
Web mapping software probably deserves several articles on its own, but as a starting point you will likely want to check out MapServer, GeoServer, and OpenLayers. For the complete 'stack' - try the OpenGeo Suite.
If you are running Windows, the easiest way to install/configure many of these packages is via the fantastic OSGeo4W installer. Simply tell the installer whick packages you want installed, and watch it download/install and configure everything for you! If you like being on the 'bleeding edge' of software development cycles, you can use OSGeo4W to install nightly development versions of many packages. This allows you to test out new features as they are developed. Keep in mind that in some cases there are bugs in these early development releases; for production work, stick to full stable release versions.
Of course there are many other software packages out there, some of which are also very good. A bit of research will turn up lots of other options! For an overwhealming list of software, take a look at http://opensourcegis.org/
Enjoy the world of free GIS!