The greatest joy in life is usually found in ( ) experiences with family and friends. Which should I choose, "shared" or "sharing"? What's the differences if both of them are right?

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@lilynyuri: no problem!
it's common structure of the lang. to see combined words in this case is "joy +of", that is why the first observation is collocation, as a tip you can learn this type of combinations by heart or with some exposure you can pick them up easily; the second issue is about articles, "the" or the "definite article" highlights that its noun easily and uniquely identified because you are not referring to "all the experiences" only to the ones "shared"; the last point is word order it's ok also to say experiences shared (sometimes rare) or shared experiences, depending on the emphasis, I modified my answer for this.😉