Birch, maple, and birch, especially those available from Home-Improvement stores, will very likely have less warp than any pine plywood.
Even better, they will have two smooth sides.
I'd be very surprised if you were not able to sort through a large pile of any of those three and find pieces straight enough for what you are doing.
With pine plywood, even paint grade, I'm not so sure you could find good stuff.
To clarify, I'm referring to plywood that is 3/4" in thickness. Many laminated plywoods are actually slightly thicker or thinner so don't be surprised if 3/4" is actually listed as 23/32" or something close.
I think the previous poster was suggesting a router to cut dados, rabbets, and miters in your plywood. These will not make it much easier for you to deal with the warpage you are seeing.
I'm sure you already looked at it, but I'll go ahead and post a picture of a large cage I built with no fancy cuts or equipment. A circular saw, straight edge, carpenters square, two clamps, cordless drill, and a counter-sink are all that is needed to build this cage.
All of the cuts were "cross-cuts" meaning the cuts were made across the width of the plywood and not the length. This means two factory edges and less complications with having un-square corners. A bit more complex for a smaller cage, though, but still easy.
A router is not needed for this cage and would only further confuse the calculations to make a flush fitting cage. Not that the math is difficult in any case, but you have indicated your are not an experienced wood worker so I think you should make this as simple as possible.
This is not the ideal way to build a cage like this, but it is completely acceptable. I built it as simply as possible to show a friend of mine just how simple cage construction can be. Throw out the false-ceiling and it would be even simpler.
