We had to replace our white picket fence panels due to insect damage, apparently the previous owner hadn’t used treated wood. When we bought the panels we were told to leave them untreated for about a month, then to paint them. While buying exterior paint I read that raw wood needed a primer before painting, but since these were “treated” I’m wondering if we can skip the primer part.

I would not skip the primer but you can ask. There should be an 800 number on your paint can. Give them a call
does “treated” mean for bugs?
then it does not mean it is primed…so call that 800# and expect to prime
Yes, you should prime the wood. I worked at Sherwin Williams for 4 summers while in college. The use of primer helps the paint adhere to the wood better. Plus, it will help the paint go farther and not just soak into the wood. You will get more legs out of a gallon of paint and primer is usually a little cheaper than paint so if you use more of that it is ok.
cedar has tanins in that will bleed through paint. You would need a primer such as Binzer Bullseye or Kilz (check to make sure it’s for cedar as they make different kinds of primers).
If you don’t prime, and properly prep the wood, the paint will peel off faster than it usually does.
If it were my fence, I’d stain it or just use a UV protectant and let the cedar weather to it’s natural color.
Pain in the butt to paint a fence, much easier to stain it.
prime before oil based paints???