How do I fix scratches from sanding drywall?
Wet Sanding
If you want a smooth surface, you could try wet sanding it.  Purchase a drywall sanding sponge (~$4.00 at any home improvement store), and use that for the final sanding pass.


Fill a bucket with water.
Dunk the sponge in the water, and then wring it out to remove as much excess water as possible.
Using a light circular motion, buff the compound to a smooth finish.

Alternatively, you could purchase this 3M drywall sanding sponge.

In has both medium and fine grade on one sponge, and it can be used wet or dry.  Similarly, use this sponge in a light circular motion.
Personally, when I'm patching drywall I tend to use a damp microfiber cloth for my finish sanding.