A quality 1/2 ply will be your best option, and there will be no need for bracing with a decent glue and screw design as long as you clamp it while drying, and use the screws correctly. By use screw correctly, I mean drill the hole in the piece you are putting the screw through larger than the screw, and put a pilot hole in the piece you are screwing into. That way you are using the screw to actually pull the pieces together, squeezing the glue into / out of the joint. Common mistake is to just drill a pilot through both (or one) pieces of wood and screw, when you do that the strength is in the screw itself, not the joint.
A quality modern glue's is exceptionally strong.
But, you will need at least 8 ply in 1/2 inch and you probably won't get that from your local Homedepot / Lows etc, you'll need to go to a lumber yard to get that.
Then once you have the lower cabinet part, I would pick the top you like, and use a glued rabbit joint to make the up rights fit into the top. That way you are using the top to stiffen everything up. 1/2" ply would also work for top, but might be nice to use 3/4.
This was a quick garage cleanup project, and the picture doesn't do the size justice, but these cabinets are all 1/2" ply, and cover the width of a 2car garage, (the toolbox is huge, 6ft wide). They have 100's of lbs of spare parts and tools in them. No bracing except the 2x2 they sits on that's bolted to the wall.
Rabbit joints are strong if you don't want to see / use screws, but to get them as strong as screw joint's / load bearing takes a lot of time. You're best off making a jig for your router to make them. I wouldn't use a dado blade in a circular saw for rabbit joints in ply. Works well on solid wood, but you really need a router for ply rabbit joints.
This is the top of an entertainment center I made that is just wood and glue with rabbit joints, not a single screw. Takes a lot of pressure (as you can see from the tiedowns) and absolutely perfect joint for it to be strong.