$A \in B$ vs. $A \subset B$ for proofs I have to prove a few different statements. 
The first is if $A \subset B$ and $B \subset C$ then prove $A \subset C$. This one is fairly straight forward, but I'm stuck on how the next one differs. 
Prove that if $A \in B$ and $B \in C$ then $A \in C$. 
I don't really understand how to put this in logical symbols. I've only seen $a \in A$ written out but never "a set $A$ is an element of a set $B$".
Here's what I have for a proof at this point, assuming I understand what "a set $A$ is an element of a set $B$" means: suppose $A \in B$ and $B \in C$. 
Then $A \in C$.

There are sets for which the relation $A\in B$  is transitive: ordinals sets, which are defined as sets on which the relation $x\in y$ is a strict well-ordering. 
Furthermore for such sets, one has $(A\in B)\Rightarrow(A\subset B)$.
Example:  $X=\bigl\{\varnothing,\{\varnothing\},\{ \varnothing,\{\varnothing\}\}\bigr\}$.
Added:
To answer directly the question, the main difference, from a logical point of view, is that, while $(A\in B)\wedge(B\in C)\Rightarrow (A\in C)$ involves individual elements-sets,  $(A\subset B)\wedge(B\subset C)\Rightarrow (A\subset C)$ involves universal quantifiers:
$$\forall x(x\in A\Rightarrow x\in B)\wedge \forall x(x\in B\Rightarrow x\in C)\Rightarrow\forall x(x\in A\Rightarrow x\in C).$$
