Choose a high-quality acrylic- or urethane-fortified latex paint (some premium exterior paints blend both urethane and acrylic.) Inexpensive exterior paint contains inexpensive, inferior ingredients and will fade and peel quickly. When buying paint, you typically get what you pay for---paint stores (not home improvement centers) carry premium paint that will last for many years if applied correctly.
Remove whatever hardware you can from the exterior door. This includes unscrewing the kick-plate, house numbers, peep hole and handle. If you can't easily remove these, use blue painter's tape to mask them after the door is sanded and cleaned.
Clean the door with detergent. Mild dishwashing liquid works well if the door is not very dirty. If there's excessive grime, use a heavy-duty detergent. Rinse it and let it dry.
Sand it with a medium-grit sanding sponge to dull the surface if the vinyl is very glossy. Wipe off the dust with a clean, lint-free rag. Tape off the door hardware you haven't removed.
Stir the paint and pour some into a clean bucket. If you work straight out of the paint can, you'll gum up the edge of the can and it will be difficult to seal when you replace the lid.
Paint the first coat on the door with a good-quality 2- or 3-inch synthetic-bristle brush or 6-inch roller. If the door had multiple raised panels or windows, it'll be easier to paint it all with a brush. For plain doors, brush around the edges first, then roll the rest of the door.
Plan on doing at least two coats of paint for the best appearance and maximum durability. Leave the door slightly ajar while it dries for about 4 hours between coats. Prevent insects from coming through by taping a sheet of plastic over the inside opening.
Replace the door hardware and remove masking tape once the paint has dried for at least 2 hours.