For corals, the treatment is a dip like Coral Revive, after which you can pretty well rely on them being clean. For zoas, which have a problem with nudibranchs (who can leave eggs behind, immune to dip) you need to qt and examine them with a strong magnifying glass to be sure they don't bring you a pest that will specialize in zoa predation. So for the one type, a dip, after which its safe to go into the display; for the other qt and observation for a couple of weeks. Revive could kill the nudis, I'm pretty sure, but until eggs hatch, Revive can't get at them. The worst plague you can get with stony corals is 'red bug,' which is a pest very very small and hard to see. Follow the instructions on the Revive bottle.
As for inverts, I rinse them off in a cup of discard tank water to be sure their water isn't bringing anything in and that nothing is riding in on them; then I just put them right on into the display. Never have had an incidence of parasites or bugs come in on an invert.