E6000--industrial strength adhesive. Look at the back. Also look at safety warnings.
Gorilla Glue--a step above liquid nails. Very strong adhesive.
Liquid Nails--a multipurpose adhesive come down from olympus. Very good for attaching building materials e.g. ceramic, wood or stone.
Mutipurpose Spray adhesive--good for wood, foam, and looots of things. Check the back.
Wood Glue--great for wood. Yep.
Hot Glue--basically magic. Multipurpose–not the best for bonding wood together (although okay for added support). Beautiful for cardboard, felt, etc.
- can be used to bond things together and create stylistic/descorative touches on a prop.
Styro Glue--good for styrofoam. Yep.
Modge Podge--works as a sealer, glue, or a finish. Check the back. Too weak for building materials.
Elmer's Glue--paper, felt, and crafty things like that.
Glue Sticks--I will trust you to use these properly. I'm sure you know what for.

Paper/Felt

You could use something stronger, but that would be overkill, and would probably have adverse effects.

Styrofoam

...

Add an appropriate amount of glue to the two surfaces to be bonded together.
Put the two surfaces together, and clamp together for the amount of time specified on the adhesive container.
- probable clamping time: 1-2 hours
- Use actual clamps
- If wood is of irregular shape, consider drilling screws in to hold prop together and simultaneously act as a clamping mechanism.
- Human clamps work as a last resort/if drilling in a screw is too dangerous for a delicate prop.
Even after clamping is done, make sure to avoid stressing joints of prop for 24 hours (check adhesive container for specific timing)