I would think the melamine stuff would be difficult to paint, but I painted my oak cabinets and they covered very nicely. My doors were solid oak and the the base cabinets were solid too, but I think the end panels were oak veneer. The end panels took the paint well too. I primed with Zinnser oil based primer and used 3 coats of BM oil impervo paint, 1 layer of oil glaze and 2 layers of poly coat (to even out the finish after wiping off the glaze.)

Here's a before and after photo

We also painted our oak cabinets and ordered new doors. We cleaned the cabs with a TSP substitute, sanded lightly, filled the grain with Behlen grain filler (2 coats) for a smoother surface, then painted 3 coats of Benjamin Moore Satin Impervo, sanding between coats.

Good luck!