I agree, 1/8" plexi should be fine, and the most economical solution to start with. Here's my process:
1) cut on a table saw. Best to use a fine tooth blade. If you have some old lauan or scrap plywood that can be used as a sacrificial layer (plexi below, sacrificial board on top) -- that's what works best for me. In lieu of that, put a strip of heavy
masking tape over the cut
line, this can reduce splintering/shattering too. Keep the saw blade as low as possible, just barely above the
thickness of the plexi, even if it doesn't cut through the sacrificial layer (though you may need to pull off
kickback/splitter or other
safety features of your particular saw -- use caution). Push slowly, don't be in a hurry.
2) A
projector and a
sharpie is the best way to draw out the
pattern, in my opinion/experience. Depending on the distance from the audience, sometimes a thick
sharpie line is all you need for the leaded glass. If you need it to be dimensional, I'll explain that step later.
3) They sell a lot of different glass paints. I've worked with el cheapo brands and they worked fine, and I've used more expensive brands and they were richer and sometimes easier to work with. My advice is have an extra or scrap piece of plexi to practice technique on. The technique I like is stippling -- it gives a nice texture, and if you have limited paint colors, you can also sort of mix on the fly. Depending on the paint, you may need to do layers (and let each dry in between) to get the saturation you're looking for. Colorations was the cheap glass paint, I believe, but not very saturated -- required multiple coats for darker colors. I don't remember the more expensive brand, but you can look on Amazon or various art supply outlets for options to fit any budget.
4) After
sharpie and glass painting is done, I've used black silicone caulk to give you dimensional leaded glass beading. You have to be very careful with this step, of course. Again, I'd say practice on a scrap piece. You might also want to consider beading on the opposite side as the glass paint, just because if you make a mistake, it might be easier to correct, but it probably would look best finished if it had glass paint and beading on the side facing the audience.
5) Be careful how you attach it to the set. I've had plexglass crack and splinter more often when attaching it than when cutting it on a saw. Even if you pre-drill pilot holes, it's easy to overdrive a screw. Best to make the windows slightly larger than the opening and use some sort of scab/frame to hold it in place.
Good luck!
PS. reading other replies, I would recommend AGAINST doing this with tape and gels (unless it is a very simple design). I've done it that way several times before, and it's just way too much trouble/time and expense, and tends to not hold up very well.