Most primers cover all sorts of materials these days.
I Recommend Ben. Moores "Fresh Start Acrylic" Know that they have two different fresh starts now. One for there top LOW AND NO Voc paints and one for there still high quality regal line of paints. Make sure you get the right one for your finish coat.
First, lightly sand and prime the top and bottom edges of the door. These are typically not painted, but can save the life of the door if exposed to moisture which is usually the case with exterior doors.
Yes, hang the door first, because you are going to scratch it while hanging it otherwise. I recommend old hinges then replace them with the new hinges after painting, because you will inevitably get paint on the hardware. Hang the handle and dead bolt if an exterior door to ensure everything lines up and works. Then take the hardware of to paint it. Leave the hinges on so the door will swing.
Lightly sand all surfaces of the doors and casings. 120 - 180 grit. 180 is best.
Start with the door closed and prime the side where you can see the hinges. Prime everything you can see, door and casing. Skip the window
You can tape it of if not able to control the brush.
Prime the edges and then swing the door open
Second: Check for runs on the other side of the door before they dry on you and paint the remaining door casing inside and out.
Third: prime the other side of the door and the check the first side for any runs again. Runs show up on the edges.
4th: Once the primer dries, putty all nail holes in the casings and lightly sand again.
5th. Caulk all gapes for air seal and aesthetics. Once caulk dries. Use quick dry caulk. I like bolt. 2-3hrs minimum, the apply two finish coats in the same order.