To keep symbology of a shapefile or geodatabase feature class, save a layer as a "layer file" (.lyr). If you keep this .lyr file in the same relative path location (in the same directory, for instance), it should continue to reference the data and you can just add the .lyr instead of the data itself. If the data and lyr become seperated you can apply the lyr to the data through the symbology tab. I've included some images to illustrate.