Repainting Exterior Doors - Grill Paint vs Appliance Paint
You do need to have a primer for sure.  

Just a tip from someone who paints 20+ metal doors a year.  Use automotive spray paint.  Cost a couple bucks more and you can pick it up at local parts store or online.  The colors are better, then finish is better, and they make some great looking clear coats.  Two coats of apple red auto paint + 2 coats of clear and I can turn a $20 door into a $500 door.  Sean is right too after your first coat you should give it a sanding.  No matter how well you think you cleaned it there will be dust that binds.  (you can also take your door to an autobody shop and most will paint it for $100 - trick of the trade is to take in 5-6 at a time for around $250 if you use common colors)