Painting kitchen cabinets - help!

Make sure they are clean and dry. Remove the doors and hardware. Mark the doors and cabinets with tape where they go. Lightly sand the doors and cabinets to remove any gloss and roughen the surface for paint with 120- to 220-grit sandpaper. Use a tack cloth or damp rag to remove dust after sanding. Prime with a stain blocking primer like Zinsser 123, KILZ,or BIN and have it tinted to the color of the top coat. This will prevent dark or stained surfaces from showing through the top coat. Acrylic, or water-base, paints are low-fume and clean up easily with water. Alkyd, or oil-base, paints require good ventilation because the paint contains solvents that can irritate your lungs and make you feel sick. Alkyd options require mineral spirits for cleanup, but they provide a hard, durable paint finish. Whichever you use, buy the best-quality paint you can afford for a lasting kitchen cabinet finish. Seal with at least three coats with a water based polyurethane. Use a small foam roller and foam brush for a smooth finish.

Use ESP then undercoat and topcoat.

You probably have Thermofoil cabinets, which have a thin layer of plastic film which was applied over particle board primarily on the door fronts. You can remove the Thermofoil plastic by carefully using a heat gun and peeling the plastic off (a 2 person job). Then you can lightly sand the cabinets, remove the dust from sanding and then prime. Then you can paint. The cabinet box that the doors are attached to are generally not covered with the plastic so you can just sand, prime and paint. Without seeing a picture but reading your description I am pretty sure it is Thermofoil. Google thermofoil cabinets and it will explain how it is applied and how to remove it. Hope this helps.

http://www.homepainterstoronto.com/blog/2014/06/19/painting-thermofoil-cabinet-doors/