What would you like to do with this data once it is in JOSM? If you plan to upload some of the data to OSM, then you first need to read these pages:
http://wiki.openstreetmap.org/wiki/Import
http://wiki.openstreetmap.org/wiki/Import/Guidelines

In particular, join the imports@ mailing list and announce what you plan to import and wait for feedback before uploading any data.

From looking at the site, the only layers which seem of use in OSM are the city limits, streets, and wetlands (and possibly landuse). For conversion from Shapefile to OSM.XML, I recommend ogr2osm, which is what I used to convert my county’s streets and parks:
http://wiki.openstreetmap.org/wiki/Ogr2osm

More possibilities can be found here:
http://wiki.openstreetmap.org/wiki/Import/Shapefile

But again, please keep in communication with the community before doing any imports, as those with more experience can offer advice on how to properly convert the files.