We, at the Institute, are big believers in free mapping software. Open Source has brought wonderful software tools to the common man who lives on a tight budget. There is no need to spend hundreds of dollars (or thousands if buy ESRI) on GIS software if you are a beginner .
The best software for the beginning and intermediate mapmaker is Quantum GIS, aka QGIS. It is intuitively easy to learn, powerful in capabilities, yet friendly in use.
Best yet, they update it constantly, and that costs you nothing. You got it — zero, zilch, nada. And as you grow as a cartographer, QGIS can keep up with you until you begin extremely advance GIS processes.
The folks at QGIS have just come out with version 1.5, and it has wonderful features. So good, in fact, that we use it for our Professional Education Training and other GIS courses. It beats out the other two contenders, GRASS, and gvSIG. You can get QGIS here at www.qgis.org