For a US county map like you displayed, a good (albeit not free) mapping package is Golden Software's Mapview. It's not a true GIS either (though it has adopted a few basic GIS functions) but is an excellent thematic mapping package for Windows. I tried MapPoint in an early release an gave it up.
http://www.goldensoftware.com/
I've used it since Ver 2. (I also like their Grapher and Surfer programs; i used Grapher, mostly, to do most of the statistical graphics for textbook on reproductive epidemiology by Allen Wilcox that came out recently. A few graphs were also done in psi-plot; another little known and relatively inexpensive stat/graphics package).
If you really want a GIS - most cost a fair bit - whether it's ArcView or Idrisi. There are some public domain ones (eg Grass). The CDC does have a free package, EpiInfo which has a GIS component.
http://www.cdc.gov/EpiInfo/maps.htm
since epi-map uses the arcview shp file format there is virtually no limit to the number of basemaps you can find.
Go to pages 29-32 to get a rough idea of what you can do with Epi-Map
http://www.epiinformatics.com/DownloadFiles/IntroToEpiInfo.PDF