Open all of the doors and windows of your workshop if you cannot work outdoors. Corian creates a great deal of dust and small particles and is difficult to clean up when enclosed indoors.
Wear safety glasses. Cover your nose and mouth with a paper mask or respirator.
Use a router saw to make large cuts in the Corian. Make smaller and finishing cuts with a trim saw. Only use carbide-tipped blades and bits when cutting Corian.
Sand Corian similarly to very hard medium-density fiberboard. Use an orbital sander with a succession of finer-grit orbital sanding discs.
Buff the Corian using rubbing compound and an electronic buffer or a drill with a buffing attachment.