This is all bad, I am sorry to say. This is the result of poor surface preparation (sanding) prior to the application of the paint that is peeling. There is no easy solution, you must remove all the peeling paint along with any that even might peel. Sanding with coarse paper (maybe a heat gun but be careful) is the likely solution. On top of that, you will need to properly sand the underlying finish, to remove any sheen, prior to repainting.
Good quality latex paint will adhere just fine to old oil-based paint IF the surface is properly prepared. No primer is needed unless you take it all the way down to bare wood.