You are describing the exact functionality of Data-Driven Pages (DDP), a built-in feature of ArcGIS. You can implement them either in a Python script or in ArcMap. If you've never used them before, I'd experiment with them in ArcMap first to get an idea of what you can do with them. From the sound of it, you should be able to do what you need without getting into Python at all, though Python lets you do some things with DDP that are impossible through the GUI.
Add the Data Driven Pages toolbar to ArcMap to get started. The basic idea is that you style your map however you want (labels, symbology, etc.), then you use one of your layers (Counties, in your case) as the "Index Layer". ArcGIS will loop through every feature (county) in your index layer, zoom/pan your map to the feature, and render a map. So if you have 10 counties in your Index Layer, you will end up with 10 maps, all with the same style but showing different counties.
There are quite a few free training resources out there to get you started:
ESRI's quick guide to what DDP does
ESRI offers a free seminar on DDP