I got some advice from my mom and it seems very sensible to me. She says she's done this many times and the trick is a fine grade sand paper then a small thin brush to apply the paint. Smaller the better as wont show brush marks. No spray paint.
She uses Latex but has no particular brand other than she's partial to Sherman Williams for exterior (Thompson's water seal for wood things).
Any other tips? I do not plan to remove the door to do this. I do plan in 2 years or so to have the door and frame removed and replaced with something nicer (It's the cheapest all white steel prehung done by renters when they kicked out the nice wood one I had).