How do I fix small chips and scratches in a melamine table top?
Scratches that are not all the way through the surface you can polish out, or sand out and leave a matte surface that will not show scratches so easily.

Chips can, at best, be filled with something that never quite matches, or you replace the table-top, as it's the only actual fix for those, IMPE. The canonical product that never quite matches is called "Seam-fil" evidently. 

The simplest effective solution is called a tablecloth.