Foam & Glue

I tried gluing 2 pieces of foam together with white glue. Even after a couple of days it had not hardened. Not sure if it was because the glue would not penetrate the foam or because there was no air circulation. Same bottle of glue works fine on a wood to wood application. I susspect that it would work on a wood to foam application. Try a test piece.

Now, when I build up some scenery with foam, I use low temp hot melt glue. Quick, clean, and no half open tubes of Liquid Nails left over.

By the way, welcome to the site.

I used LATEX contact cement to bond my foam sheets to each other and to the plywood,  holds really really well. 

Be sure it is LATEX water clean up contact cement,  regular contact cement will disolve your foam into a puddle of goop  (my brother didn't think I meant it when I told him ONLY Latex and he not only desolved his foam, but it ended up ruining a carpet under it.).

I have tried several other types of glue subsequently to hold bits of foam,  liquid nails is great.  latex chaulking is good.  In a pinch I have used yellow carpenters glue (variation on white glue), it is just OK and only if you leave it a week or so and it is exposed to air,  it is not good laminating foam sheets to each other or to plywood. 

Hot melt is good for a quick fix, but is a brittle bond.  tend to avoid that approach. 

Have used Gorilla Glue and other clear type glues,  they do react to the foam, but in small bits can old, eg, gluing a tree in place, but sometimes a bit of goop foams where it is applied,  avoid glues with solvents in them.

my two cents worth on this topic from my experience.

You could try polyurethane wood glue. It is similar to polyurethane foam: it expands when it hardenes so it fills small cavities easily and it absorbs moisture. Just remember to wet your foam before applying it and secure the pieces well. I used it to bond wood to foam and it worked well except that the stuff is rather expensive and messy.