What are you trying to do specifically? Do you need the software for projecting, viewing, and manipulating information you already have?
or are you looking for something with the state and county data included?
Do you in fact just need Images of county maps at different scales?
The answer depends heavily on what you have available/have access to and the final product you want.
If you are a student, the first place you should go is the geography department. Ask around and you may find there is a GIS lab for you to use. After running a couple tutorials (one or more of which might even involve US-scale mapping at the county level) you should be able to pound your way through making maps. OTOH, If there are any students working in the lab they can probably make a basic map for you in 5 minutes. It’s been about ten years since I used any kind of GIS tutorial, but I seem to recall a US county map in an ArcView tutorial.
I use pretty advanced GIS software at work, but when we do work for non-profits, etc. we can distribute a freeware application called ESRI ArcExplorer that allows them to use information we provide:
http://www.esri.com/software/arcexplorer/
This software will allow you to make and edit maps, provided you have the data in a format it can use, e.g., “shape files” These files are often freely available from Government agencies and Universities. For example, If you had a US-by-counties shapefile, you could use ArcExplorer to make basically any map you want.
For US stuff I would check the USGS, Dept of Census, etc. I did a quick Google search for “USCounties.shp” and found this site at Iowa State:
http://mesonet.agron.iastate.edu/data/gis/shape/unproj/us/
I did not try to download the files, though, because I’m not sure it would be legal for a non-student to do so. Mods please remove the link if this violates board rules.
Of course, this is way too complicated if all you really need is images of certain counties at certain scales. That kind of thing can probably be found on the internet by searching Google images. You could then Edit the image to display what you need. This’d be really time consuming to scale and assemble a composite image of the entire U.S., though.
I can probably give more help if you explain what exactly you’re trying to do.