I have used a low-abrasion toothpaste and a lot of running water with good results. When I get a DVD that has grease or something else on it that makes it skip, that will generally cure the problem.

I wipe CDs on my shirt.

If the problem is worse, @CyanoticWasp has your answer – toothpaste. It polishes plastic very well. Works great on clear plastic watches with smudges & scratches.

Wipe the cd in the radial direction, not tangentially. In other words, back-and-forth between the hub hole and the outer edge, but never in a circular, concentric direction. (This is to minimize the time duration of scratches so the error-correcting software can handle it.) I use some combination of soap & water and alcohol, wiping with cloth rather than paper products.

@CyanoticWasp Do you have a particular brand (or even better, exact product) that is low abrasion?

@papayalily pretty much any gel product would work, I think. You want the plainest, most ordinary toothpaste you can find, basically.

Preferably, use a piece of cloth that is 100% cotton, or try to see if you can get a piece of cloth from your optician that would be used to clean spectacle lenses.

I use warm water, soap and my fingers. Then I dry it with a soft cloth, but not in a circular motion. That always works for me.