Can it get wet? It depends on porosity and composition
This is where most mistakes happen, because instinct says "it's a stone, it can handle water." But there are three quite different groups.
Minerals that tolerate water well: quartz in all its varieties (clear, amethyst, citrine, rose quartz, tiger's eye), agate, jasper, obsidian, garnet, topaz. A damp cloth or a brief rinse with lukewarm water won't affect them.
Porous or soft minerals, best avoided or only briefly exposed to water: turquoise, malachite, larimar, opal, lapis lazuli, sodalite, azurite. They're porous or contain inclusions that can absorb moisture, lose color or develop micro-cracks over time. They clean better with a dry or lightly dampened cloth, never submerged.
Minerals that water can directly damage or dissolve: selenite and gypsum (can eventually dissolve with prolonged exposure), halite/salt (dissolves literally), pyrite (water can promote its oxidation over time). With these, never use water — only a dry cloth or a soft brush for dust.
In short: when in doubt, lukewarm water and a soft cloth are almost always safe for hard, compact stones; for everything else, a dry cloth is best until you've confirmed which mineral it is.