I wouldn’t like to use plexiglas as it really limits where you can put toys and perches. If you really want plexiglas I would personally use it on one side of the cage, and if iT is to keep the mess in, I think I would try to design a sort of system where you could place the plexiglas over the mesh, and remove them when you want to clean.
However no matter what you decide, I think just regular plexiglas is safe? The transparant regular kind? I am not an expert though. As for the mesh, I would choose for Stainless steel, iT doesn’t rust and is completly safe. And the wood, well there are a lot of safe wood types, but I would make sure they can’t get to the wood, otherwise they may start eating away at their cage! I guess you could simply Solve that by screwing an extra piece of wood on the poles on the inside of the cage, so they Will chew that and not the actual frame of the cage, and when they have destroyed that piece of wood you can screw on a new piece.