I use 6mm (1/4") thick fun foam, and cut it just a teeny bit wider than the stamp so there is something to grip when inking up and stamping.
The glue is some white, general purpose glue* that is allowed to dry to tacky before touching the pieces together. You can still peel apart and adjust them for about 15 seconds.
I score the back of the foam with a knife point, and the back of the stamp in the opposite direction with a small gouge, and give a little twist as I smack them together. It has to dry about 4 or 5 hours before you can stamp or it may slide sideways a little. It depends on how long you wait to put the pieces together, longer being better.

*Crafters Pick from Hobby Lobby - water based, non-toxic, and seems to resist water when dry, since I have successfully washed stamps in the sink. I don't soak them in the water, but run a trickle across the stamp as I clean it, so I guess it's OK.

I get the thick foam from Stampeaz for $ .69 a sheet. She has lots of colors.
Edit: I don't use the foam on the new Japanese carving material. It's thick enough on its own.

I am grateful to anyone who mounts their stamps with some kind of backing because I tend to drop them ~ OFTEN!

I like to mount my stamps on 1/2 foam, I found a piece of foam laying by the road to my home, it is that alphabet letter, square, that is for a floor. It is flat on the bottom and textured on the top, I use lock-tite glue that seems to work really well. I always cut it to fit the stamp..with very little overhang.

Thank you everyone! This is super helpful! I didn't think to put foam between the stamp and wood block if I go that route. I'll be going to the craft store this weekend! :)