Plan to use an alkyd or good-quality latex paint over a compatible primer. Talk to your paint dealer about the best choices for your cabinets. Make sure that the primer is suitable for use on melamine. Semi-gloss paint would be a good choice.

If you are using latex, consider mixing it with some paint conditioner. Latex dries quickly, which results in more prominent brush strokes in the finished surface. A good conditioner will lengthen the drying time some, giving the paint more of an opportunity to level out. This difference in drying time is one reason many painters prefer to use alkyd paint for trim and nice woodwork, and you might want to do the same.

Another way to ensure a smooth finish is to use a good-quality paintbrush. I would not use a paint roller or foam brushes for this type of work.

Apply one coat of primer, then lightly sand and wipe the surface again with a tack cloth. Apply several thin coats of paint, sanding in between coats as necessary to remove any dust or debris. Allow the paint to dry thoroughly.