How can I stop the cut edge from looking chippy?

In addition to other answers re use of painters tape and cutting from back side it also really helps if you use only enough of the blade to cut through the material your cutting. You don’t need to expose 2” of blade to cut a ¼” thick piece of wood

When using a circular saw or jig saw, you should cut with the saw on the BACK of the panel. Both of these blades cut on the upstroke, so by cutting from the back side, any chipping or tear-out happens on the back, not the front. Also, it's a good idea to tape the front along the cut as suggested above. For cutting on a table saw, you would cut the panel with the good side up, and the back side down. Another option is to score the cut line with a utility knife on the good side, then cut along the score line. The score line should help keep the finish from splintering.

I always tape the area I am sawing ON ANY MATERIAL with masking or painters tape to avoid this problem.