There are many situations in which we would wish someone "Good luck!", for instance when someone is embarking in something difficult or dangerous. How can we express a similar thing in Italian?
The "neutral" way is the literal translation:
The more colloquial way (not rude or offensive, can generally be used with anyone, though it might depend on the situation) is an idiomatic expression:
In bocca al lupo!
to which the person who is wished luck usually replies:
Crepi (il lupo)!
I'll depart from the obvious answers, and suggest you to say: "Auguri!".
This is the most neutral greeting in most cases: no particular register, doesn't assume a good or bad occasion, doesn't invoke the sorte etc. Can sometimes be sarcastic; but if you are sincere it won't, just like "Good luck".
I'm pointing this alternative out because it happens to me almost daily that I think "Auguri!" for someone who only speaks English, but I'm prevented from saying it because in English any translation may be ambiguous.