Is the place to download map tiles, down to street level, in a zip file by county, state or region.
Not as far as I know. However, if you download map tiles on one machine, you can simply copy the resultant directory tree onto another machine as long as you have the tile set configured to point to the proper directory root.
Oh, and I should say that APRSISCE/32 doesn't do it's own maps, but is a map tile client accessing OpenStreetMap-compatible tile servers to get map tiles for caching and display. See also http://aprsisce.wikidot.com/tile-sets
The web-site GeoFabrik has files by state. Which format, if any, should I download?
As stated at the wiki page mentioned above, APRSISCE/32 only supports OSM-compatible map tiles named the same as they name them. Check the directory tree for one of your map tile sets and you'll see the files that need to be available for the maps to be displayed by APRSISCE/32. I suspect that GeoFabrik doesn't support that particular format in their downloads.
The Geofabrik maps are OSM. They just allow you to pull maps by geographical area or they have already done so. They list a few different compressed files to download, I just couldn't figure while one I needed for the selected state.
Geofabrik maps are OSM data, not OSM-compatible map tile. They offer downloads of the raw OSM data (extracted, so you don't have to download the entire planet) and shapefiles. The OMS data is what you'd feed into a map server that would render tiles from that data. Shapefiles are supported by some mapping software, but not APRSISCE/32.