"Good luck with that" is a set phrase that people say when you're going to try something that they think will be hard or impossible.
The literal meaning of "good luck with that" is simply, "I hope that you succeed with that." However, this phrase has become popular to use sarcastically. So it often means "I don't think that you're going to succeed."
Any kind of sarcasm should be used carefully, and only with people who you're close with.
If you sincerely want to wish someone good luck, you can simply replace "that" with a longer phrase:
Good luck with the competition!
You can also use "on" instead of "with":
Good luck on your exams.