Remove the doors from the cabinets by unscrewing the hinges. Number the doors and place a corresponding number inside the cabinet where the door belongs. This will eliminate the possibility of replacing the doors in the wrong order later. Use painter's tape to create a numbered label that will not cause a permanent mark.
Remove the handles and pulls from the cabinet doors.
Clean the doors with a grease-cutting cleanser, such as TSP. Because kitchen cabinets are exposed to years of smoke and hands covered in cooking oils, it is particularly important to make sure to do a thorough cleaning job--paint will not stick to a grimy surface. Rinse the soap residue from the cabinets and allow them to dry.
Examine the doors for chipped veneer. Fill any chips or holes with an epoxy filler.
Sand the doors with a #120 or #220 grit sand paper. Sanding the surface will slightly rough up the surface and give the paint something to grip onto--veneer is smooth and not likely to adhere well to paint without sanding.
Use a tack cloth or damp rag to remove the dust particles that are created during sanding.
Prime the doors with a fast-drying oil-based primer. Use a roller to avoid leaving brush strokes. Allow the primer to dry fully; use the recommendation of the manufacturer to determine the primer's dry time.
Apply an oil-based paint or enamel paint designed for cabinetry. Water-based paints can also be used, but will often take much longer to cure completely. Use a roller to apply the paint so that you won't see brush strokes. Allow the paint to dry for at least two hours and then add a second coat. Once the paint is dry, reattach handles and replace the doors.