Walter, I am a certified fabricator of Corian and have been so so a loooong time. First make sure you have Corian and that it is 1/2" thick material.
Some of the galley tops are a thin sprayed on finish and called 'Corian'.
Foileaduex is right on and his instructions will get you a pretty good semi gloss surface.
We skip some of his work, what you need to know is how deep the scratches are and start with the appropriate 'grit'.
My guess is that with normal wear you could start with 180 or 220, but back it up with a soft pad between the paper and the random orbital sander.
We then go to a purple scotch brite, then grey, then white. (different colors are different grits)
This might speed up your process.
And if not you can always go back a step and start over.
The only comment I can make on his suggestions is that Corian is none porous so waxing is not necessary.
But what the heck if it works for him then, 'there you go'.
Lighter colors can be restored without a lot of fine grit finishing.
Darker colors will need more effort and will require that the finish is more glossy.
However I strongly agree with his suggestion to practice first.
Call around and find a scrap to practice on.
Or go to
http://www2.dupont.com/Surfaces/en_US/assets/pdf/Corian(R)_Care_and_Maintenance.pdf
Good luck and it's not tough.