You could copy the county polygons to a separate "mask" shapefile. You may or may not have to merge the polygons to make a "mask". Using this mask shapefile you can clip each of the different shapefiles to make new shapefiles for those that are just in the three counties. The tool for this is Analysis Tools.tbx\Extract\Clip. For rasters data you can use Spatial Analyst Tools.tbx\Extraction\Extract by Mask.
If you want to use the symbology from the original shapefiles, you can right click on each of the original shapefiles in Arc Map and choose "Save as Layer File". This creates a layer file which stores the property settings (symbology, labeling, etc.). To make this apply to your clipped shape files, bring the layer files in to your mxd (it will have the source of the original layer) and change the source for each one to be the clipped shapefiles.
There may be a tool that someone has made that will do all of this in one tool, but I don't know of one.
Another option would be to use a function in the Data Frame properties (right click the data frame and find at bottom of menu). Under the Data Frame tab, at the bottom, there are clipping options. This allows you to use a shapefile to clip the data without actually having to do all the stuff I mention above. There are problems with getting this to print on paper the way it looks on screen so before you do to much work, make sure it prints out as you want it with the clipped data frame.
One last option is to copy the state polygon and make the mask of the three counties as mentioned in paragraph 1 above. Then use the Analysis Tools.tbx\Overlay\Union tool on these shapefiles. Then delete the features in the shape file defined by the "mask". What you have is a reverse mask that you can lay over the rest of the data to mask it out. Edit the mask to expand it's outer edge as needed. This will do some of what the data frame mask does. You may have to deal with labels of features outside of the county boundaries though. Depends on how much labeling you plan on.
