So, let me expand on the title.
Suppose a set $X$ is transitive, is it then true that $\in$ is a transitive relation on $X$?
I'm just slightly confused by what the question is.
Assuming $X$ is transitive, then $(a\in b\wedge b\in X) \implies a\in X$. But, is the question above asking if for $a,b,c \in X$ we have $(a\in b \wedge b\in c \implies a\in c)$? This just means that $c$ is itself transitive, and clearly this is false since not all elements of a transitive set is transitive itself.
The other interpretation, meaning $c=X$, seems a bit easy since it's literally the definition.
I know it can be difficult to deduce the author's meaning of a question, but I'm hoping some of you experienced folk know the intention of this question.
Your first interpretation is the correct one. The question is asking whether $\in$, considered as a binary relation on the set $X$, is a transitive relation. You are correct that the answer is no because an element of a transitive set need not be transitive (though for completeness you should perhaps give an explicit example of this).