Handling corals will really depend on the type of coral. My opinion is that it's best to always handle them with rubber/latex gloves, just to be safe. Everyone has different allergies and sensitivities to chemicals, and there's really no way to know how someone will react to a coral "sting". Then again, it's not often I do that myself anymore.
Zoanthids.... ALWAYS handle with some type of protection - Google "palytoxin" and you'll understand.
There's also different ways of actually holding/handling them, but that depends on the type of coral. Best thing is to ask your LFS
, or post up a specific type of coral here for advice. A generic bit of advice for LPS
corals... don't touch the living tissue, and try and hold it by the exposed skeleton.
Can't help ya with anemones... never had one.