Seriously consider a trip to a pro paint shop...
I'm not a big fan of Glidden gripper, I'm guessing you bought it at Home Depot or the like.
That said, it WILL do the job, but spend as much as you dare on your topcoat then. Latex Enamel will work fine but you need a paint with lots of solids and binders. That generally means expensive paint.
Alkyd (Oil) enamel will work very well as a topcoat too, but will be very expensive. It does leave a superior finish. Your best job would come from
Shellac sealer, oil primer and topcoat. But it would be timeconsuming, expensive, and stinky.
If you really want the finish to last and not chip (the main problem with painting melamine) scuff the melamine with 150 sandpaper or use a liquid deglosser, wipe it CLEAN, and use a *better* bonding primer like Benjamin Moore or Sherwin Williams PrepRite or another top-end product. Consider adding a little flow-conditioner ("Flotrol") to your topcoat paint to smooth the aplication of the finish.
At the risk of sounding snotty--you will save a lot of time and frustration
getting the advice of a pro-grade store if you have one in your area.
Worst case: you'll have a finish that mars, is perpetually sticky, and suffers other annoyances.
Sometimes local small hardware stores can have pretty smart paint folks too.
I'm sure you want the job to look great, and better grade paints make that easier to do.