Doors, frames, and casework are all considered trim work. They are usually constructed of wood, vinyl, or metal.
Because doors are the entrance and exit points, plan to paint them last, so the paint finishes will be undisturbed and allowed to dry upon completion of the room.
The easiest way to paint a door is while it is in its frame, on its hinges. Clean the hinges with rubbing alcohol, then mask them with two coats of rubber cement (peel the cement off when finished). Prep the same as window trim. Remove or mask the doorknobs, lock, and other hardware.
For any type of door, start by painting the frame (casing), working up from the inside bottom, across the header, and down the striker side.
To paint a flat door, start by painting the inside hinge edge, working around the door in one direction. Use a combination of rolling and brushing, applying the paint with a 4-inch closed-end foam roller. Run two or three roller widths the full height and across the door face, then lay off the finish by brushing from bottom to top with a lightly loaded brush. This technique allows the roller to deliver the paint quickly and evenly to the surface while keeping a wet edge, and leaves a smooth brush finish.
Your approach is a bit different to paint a paneled door. Apply the paint with same roller and brush techniques. Begin by painting each panel, starting with the upper left-hand panel working down the door face in sequences. Starting from the bottom of each of the center vertical stiles, lay down and brush out the paint. Next, working from the top member down, continue painting each horizontal member. Finally, paint the full-height outer stiles and edge. Lay off any runs or sags as you paint.
Allow the paint to dry, lightly sand, and apply the second coat. When the paint is dry, score around the edge of the hinges with a knife and peel away the cement. Replace the hardware.
Keep paint off the floor by sliding a piece of cardboard underneath the door. Brace the door with wood shims to hold it steady.