Hi,
i did a few of my own display case using plexi-glass/perspex/acrylic plastic (they all the same..)
the best solution for the 'glue' is chloroform. This thinned solution works best and since it's really thin the capilary action will work itself into the gap between the two joint surfaces. BE carefull not to let chloroform runs out of the joint as it will leaves marks on the surface of the clear plastic.
You got a lot of good respond here..but i may add that when working with plexy glass, do not completely remove the backing paper. Just tear about 1-3 cm along the edge.
and finally, u may apply any color tape along the joint to further enhances the display case. (this is optional..)
here's an example of display base i made for my MiG-29.

for the wooden base, i use 2mm plywood cramp together with 1.5" wooden frame inside the cramp plywood. and then i used decorative wood borders to the side.

hope this help!
errr...i didnt coat it... just clean like you normally clean your glass....