First, do a good cleaning on the door using warm water and a little dish soap. Now you need to sand the door using 150 grit sandpaper. Try to do some extra sanding on the damaged areas, feather out these areas to help hide them after painting.
Remove the dust and apply primer to the entire door. Now you can paint. Since primer was applied you can use any type of paint. A good acrylic semi-gloss trim paint will do a good job.