I would use a good quality primer on the door. But you still need to scuff it up and clean it well before even painting the primer on. What is normally on the door is a factory prime to prevent it from rusting.
We had scuffed it up & painted without using a primer, but the brush marks were pronounced. The sales rep @ Home Depot said it was likely my bush causing this & said the primer wouldn't make a difference with the streaking.
He is correct on that. It is not easy to brush a door without marks. Your best thing to do is wait for it to dry real good, several days. Then sand out the brush marks, Cut the door panels with a brush then using a smooth roller do the flats to remove any of the old brush marks. You do not have to prime again.
Scott, you only need to prime if the door is rusted. If it's exposed to getting wet, check for spot rust around the bottom.
In addition to what Woodbridge mentions to not have a problem with brush strokes showing, here are some more hints:
-get a good Purdy brush
-use the best quality paint from a reputable paint store
-oil-base paint looks smoother than latex
-paint in the cool of the day
-you must work quickly to keep a 'wet edge' everywhere
Also, here is a post on how to protect the vulnerable bottom edge from rusting
New metal doors generally only need to washed first then painted with the coatings reccommended on the finishing label found on either end. Many say to use acrylic paints only since the primer used isn't compatible with oil-based alkyd paints.
If the paint is fairly new then let it cure out for several weeks before sanding out the brush marks. Use 3M's green paper since it cuts cooler and doesn't clog up as fast as the others. Add a drying retarder like Flood's Flo-trol to acrylics or Penetrol for alkyds so the paint will level off after application. On paneled doors you do the inside the panels first, the short stiles between the panels second, the longer stiles third then the long ends last. Or do like me and work fast from the top to bottom doing everything along the way so the paint doesn't have time to set-up. I don't bother to ensure one-coat coverage since 2 thinner coats are easier to apply plus dry faster. It also helps to have the lockset and dead bolt removed during painting. If you need to close the door after each coat then most weather stripping on the frame can be removed first then reinserted once done.