One beginners mechanical project is the cardboard automata. This lesson plan comes from the Exploratorium PIE:
This site specializes in directions for "paper animations" that include mechanical devices (cams, levers, gears, and more). You print the templates on card stock, cut them out, and glue them together. I (Gretchen) purchased a $14.95 membership, which gives me access to many templates for free. Let me know if you want me to get something for you.
You can get the craft foam ("Foamies") at Michaels. You may be able to use a cut up styrofoam plate, I haven't tried it, but it might work.