Yeah here i my take on this, for the bryce screen illusion I would suggest Masonite or quarter inch ply wood. The difference between the two is this, one is more like card board and can be cut with an utility knife; the other will need a saw.
To attach the panels together, I would using gaffers tape. (basically its a very strong fabric tape.) Its the stuff used for the box in "Origami" Jim Steinmeyer. You may be able to find it at theater supply stores or sound equipment store. If that is not available I suppose you could get away with Duct tape.
Just make sure when you are taping the panels together to leave about 1/4 inch gap or more between panels so that they can fold into each other. Another tip is when you are taping the panels together, don't tear off a big strip and try to stick it down strait. Grab a friend and slowly unroll and stick.