I would use Fine Paints of Europe for a beautiful door. I think they can help you get the best finish.

Clean the door and frame well, sand it down good and feather out all areas where it is peeling. Clean the door, the frame and all its surroundings good, and then Prime it with a top quality exterior primer, and then paint it.

Thanks all for your helpful replies. For the existing peeing paint, should I use a paint eater tool? Hot air gun? I am trying to understand if I should remove the entire existing paint before applying new paint? Or should I only clean the peeling part and leave the other parts there? If latter, then the area wont be smooth even with sanding because there are areas where the paint is peeled off and some areas will have old paint. Hope it makes sense

I have replaced the old door that you see in the photo but the frame and case need to be painted.

You do not need to remove all the paint, if it is adhered well. Where it is missing, you do need to sand well, feathering it in, so that the peeled spots wont show once you are done painting.