We install a lot of Corian tops at work. After the seams are epoxied together they are sanded using progressively finer sand paper. 320 grit sandpaper in a random orbit sander (or by hand) should polish them up nicely. If you really wanted to do it right you could start with 150 grit then 220 then 320. I would hit it with 320 first and see if that does the trick. The finer the paper, the higher the shine you'll achieve.
Corian (Hanex, etc) look really nice high polished but the down side it it show the scratches. Keeping a slightly dull finish (220), although not as shiny and pretty, is more user friendly.
John