I used
3/
16" plywood on my Pontiac. It's stiff, but not too stiff, and it's fairly easy to work
with. Also it takes staples well. 4 mm staples in my case.

When I got the holes for the crank and latch handle drilled I put the panel on the door and drilled holes for
generic plastic interior retainers. The ones I used had a big flat head, and the stem had like thin fins to keep
them in a hole. Like this:

So as the interior panel was held in its place, I drilled the holes through the plywood and into the door. Removed
the panel, then pushed the retainers into the plywood, stapled the door covering material on the panel, and then
just had to aim the retainers into the holes in the door, and give every retainer a bump with my fist. Very
straightforward, and no visible screws or such. I assume that kind of retainers can be found at your auto parts shop.