I'm a total newb here, but I have a background in making foam LARP weapons that I think can give some insight. I'm going to walk through a normal weapon build, hopefully some of this can be extrapolated and applied to foam armor techniques.
When I build LARP weapons, I generally start with a
Microcell Foam like you see here, which is formed around a 505. or .524 kitespar/fiberglass rod for the weapon's shaft (not relevant to armor here).
You can form the basic foam shape with a band saw. Since I don't own a bandsaw, I often resort to using just the blade of a hacksaw, which works just as well if you've got a steady hand and good eyes.
If I want to indent into the foam, I use either a dremel router attachment (very good results) or a wood-burning tool for simple indented lines (once again need a steady hand and good eyes.)
If I want to build up upon the foam with raised details, I tend to use the cheap funfoam sheets you can find at Walmart or any craft store. (these will be thin sheets of multi-color foam). You can also find big 2 yard rolls of the stuff at JoAnn's Fabrics if I'm not mistaken. (good for coating a larger surface with no join seams.) This kind of thin foam I tend to cut with scissors or a razor, both work equally well.
When I need to connect pieces of foam, I use
DAP Weldwood OCC, affectionately known to me and some LARPer craftsmen as "Ogre Snot". It produces some pretty nasty fumes so a respirator, or at least working in a well-ventilated area is recommended. This stuff will stink up your workshop so it's best used outside with a box fan. The reason DAP is used is because these weapons are full-contact and hot glue just wouldn't hold up to that kind of abuse. The beauty of DAP as a foam glue is that it's relatively cheap, it expands just enough to fill gaps, and when it dries it's pretty much waterproof. I apply DAP with a spare trangle scrap of foam, apply very thin but fully to both edges you intend to join. Within a minute or two the DAP will become tacky, at which point you can join the foam edges. The edges will stay together and you can continue working, but it takes about 12 hours at least before the DAP dries completely. The result is a bond that's nearly unbreakable, the foam is in fact weaker than the DAPed seams (like a weld).
If you do it right the joins should be seamless, but in case they are not I've discovered that
Tulip brand Puffy Paints fill in these wider gaps pretty well. You can then use very fine sanding to make everything smooth.
When all the initial design is set in foam (be sure the foam is as smooth as possible, all embossing details like battle damage included), I spray on PlastiDip spray as a first coat, then use the paint-on Plastidip as a second coat. The reasoning behind this is the spray-on PD goes on very very thin and doesn't soak as deeply into the foam. It's like a primer to prevent the thicker PD paint from it's normal habit of soaking as deeply into the foam, which is just wasteful. It also makes the PD Paint results smoother. This should form a tough but flexible rubber skin over the foam.
Next step is painting. I use acrylic paints because they have the best results I've seen for flexible PD surfaces. Spraypaints tend to flake and chip off, and enamel/oil paints tend to react badly with certain kinds of sealers. (had a problem with Krylon Clear Coat)
After that's finished, you're nearly at the finish line. You just need to seal in the paint to prevent water damage and paint runs. I've seen some LARP crafters swear by Krylon Clear Coat, but I've had problems with that stuff flaking off. I personally prefer
Thru The Roof sealant, although it is tough to find in stores (I have to internet order the stuff.) The reason being that TTR will form a flexible rubberized clear coat, arguably stronger than the PD itself. This is a clear roof sealant, when you open the can you'll notice it's very very thick, almost taffy-like. You need to mix it with Mineral Oil to make it similar to pancake batter in consistency, no math here just keep mixing in a little bit of the mineral oil at a time and stirring vigorously until it feels right. This stuff paints on, I tend to use a thick 2" paintbrush (just makes sure your brush isn't losing hairs before painting, it'll end up embedded in the clear coat) but you can try other ways to apply it. With the Thru The Roof coat wet, WYSIWYG and the finished result looks very similar to the wet results so make sure you get it the way you want. A word of warning, you'll want to apply and move on, spend too much time brushing it in one area and it will glob up on itself and look bad. This is something you'll want to test on a scrap piece a few times just to be sure you get the results you want BEFORE applying it to your armor. I honestly get the best results with the stuff when I nearly drybrush it on, a very thin coat will do. The thicker the coat, the more glossy it will appear so keep that in mind. The mineral oil mix will make this stuff take forever to dry, I like to give it 48 hours, but you may need even longer if you go for a thicker coat. You'll know when it's finished because it stays very tacky to the touch while it's still drying. Make sure not to leave it in a dusty area, or anywhere near pet hair.
That's pretty much my method. Hopefully this read, while off-topic for Armor, might help with the Foam side of the construction.