Can I Use Polystyrene Foam Or Wood As A Core?

Yes you can; but when considering wood read this first:

WOOD

Crushed tinfoil is always the first best option. It is absolutely waterproof, has terrific grip, is commonly available and pretty inexpensive, which is why we absolutely recommend it first. Wood can be used as an armature but you need to take the following into consideration:

As wood is porous and flexible and Pal Tiya Premium is rigid and dense, if the wood distorts at all it can put enormous pressure on the Pal Tiya Premium. Wood swells in the presence of water, especially in the curing/strengthening stage - this swelling can apply pressure inside a piece and cause it to crack.

Ensure that any expected distortion in the wood is prevented from applying any force against the Pal Tiya Premium shell. 

FOAM

With foam, the short answer is paint the foam first, dry thoroughly, then apply foil. An outdoor Acrylic-based paint can help seal it. 

White bead foam is the exception, hot glue sticks lightly to this surface.  This would only become a problem on a very large vertical piece, such as an adult size standing human figure.  

The solution that is perfect for a project and will get the best possible results is paint first in acrylic to seal the foam, dry it thoroughly, then use foil and hot glue. 

Also, make sure that there is enough water/moisture present at least during your creation's first 5 days of curing. Remember, keep it wet wet wet.

We have a comprehensive video on Armatures for all sculptures here

We have a shorter one here