Primer's purpose is helping the color coat(s) stick.
Zinc in the primer is there to aid in corrosion resistance.
Pretty commonly used in the US Navy... we had several primers that had zinc. We never left the primer as the final coat, whether it be on the hull, piping, airplane... whatever. (Yep at a certain point in painting a F-18, its Zinc Chromate green)
Just the name "primer" points out its intended to have other paint applied after.
You just need to dust off the primer with a clean dry rag if you haven't been handling the part. if you've been handling it you should use a solvent that will not attack the primer to get the oils from your hands off of the paint.
Look on the can directions or get the MSDS from the store where you bought the paint. the paint maker will tell you what solvent to use if you need to degrease the primer.
Handle the parts with nytrile or other chem resistant gloves between coats and avoid needing the solvent...