First thing you should do is hit the door with a Scotch brite pad or some 120 or 220 sandpaper, until you’ve scuffed the entire door up.
Definately always use primer, it’s much cheaper than paint, and allows you to use less paint while getting fuller coverage. Without primer, the tone of the wood grain would show through in certain lighting, but 3 coats of primer will give you a solid color canvas to work with, and make your red more vibrant.
Exterior latex paint should do the trick. If you’re worried about chipping or cracking, do a few dozen layers of spray on crystal clear after your paint, scuffing with scotch brite pads between all coats.