Is there a way to export a shapefile to Google Earth format (*.kmz) using QGIS?
QGIS will not save/export straight to KMZ. As Aaron suggests in his comment above, you can right click on your layer in QGIS, select 'Save As...' and choose KML.
To then convert your KML to a KMZ, you have two options:
Open the KML in Google Earth, right click on 'Save Place As...', and select KMZ as your desired format.
Put your KML in a .zip file, then change the file extension to .kmz. You will now have a working KMZ file you can open in Google Earth as normal.
Check this plugin out. It exports also the symbols as displayed in a QGIS layer: