I have used the foam board before.  I believe it is called TYPAR and is used for home insulation. It comes in 4 x 8 sheets. You can buy it at Home depot, Menards, etc. It is not expensive. It is very lightwieght, can be cut with a razorblade, you can paint it. Only one negative thing about it.  It breaks very easy, so if you can attach it to a frame, or permanently on a back wall (for the whole show) it will be fine. But if it is going to be moved on and off alot, it will probably not last long.  You will find yourself tapeing and repairing, etc. Good luck!

 

I see the TV studios are using the sandwiched foam boards as flats these days, Which would have to be better than the cardboard & sticky tape, they used in my day.
Being very old [ancient]. I still prefer good old calico/muslin & pine frame flats.
TV flats work & they are easy to produce, but as suggested are susceptible to damage & also become difficult to repair effectively, with constant reuse.
But that?s another topic which has been raised here a few times.
However if you use hard flats, you can always skin over them with unbleached calico to give a Jesso effect & have the best of both worlds, making the skin surface easier to repair & with foam board it would reduce structural damage from scene movements.
In this case I would still use unbleached calico being raw cotton fibre, after being wet it will shrink & really tighten up over the board surface.
With the frame you could just butt joint & even use the gang nail joining plates. Rather than just relying on gluing the frame to the boards.
as it seems your only planning a Gaff wall of flats, bracing is still needed, as you would possibly use about 3 flats. Adjustable stage braces or the French bracing is easier to make up. Also look at a drop on ledger rail brace, 10? in length.
With the French brace & joining the flats together. Removable drop in pin butt hinges are a good option. If they are difficult to find, stick to a standard off the shelf butt hinge available.
It is better to get the type with an even number of pin captivation bearing holders, 4 is great, but 5, with 2 on one half & 3 on the other can be a pain.
Just grind off one end of the pin & remove it leaving the hinge in two halves. Buy a length of steel rod the same diameter, cut it into sizes at least double the length of the pin size. The extra length will give enough rod to roll/bend into a finger grip at the end of the pin.
Grind & chamfer the insert end of the rod, to make it easy to insert the pin. Use some white cord [easier to see in the dark] tied to the rolled finger grip end & secure the cord at the hinge half on the flats side rail. This way there is always 2 for each hinge & there is a spare on hand to drop in place.
You can make up a template with half hinges on just a spare bit of timber, so the hinge halves will dovetail together the same on each side rail of the flats.
When fitting the French braces in place use the same system. Leaving the hinge halves, for future fit up of the braces, with the pins it is easier to attach them to the braces, the same with rostra, treads or set pieces.
Just tec screw the halves on as the screws can loosen to give a bit of wiggle room when bumping in/out.
However with soft flats I prefer to use a sash cord & cleats to tighten them together. Which makes it much easier with just a flick of the wrist to serpentine lace them up, especial for 18 ? or higher  flats. It is also quicker to bump out just to untie the sash cord, allowing the flats to fall & float down on a cushion of air, before they actually hit the stage deck.
Although on bumping in a season standing set, I use the sash to place the flats & the drop pin hinges as well!
 

In the past we've used the foam-type board and, by creating a "sandwich" of sorts, easily made it stand.  Frame out the foam, first.  Then, screw shims (3 on either side) perpendicular to the foam and attach your second sheet of framed-out foam to the other end of the shim.  You create a sandwich of air between the foam panals.  It's really easy for a teenager to step into the sandwich, pick it up, and turn it around to change the scene to the other side.  By removing the shims, you can store these flat.