Do all groups implicitly act on a set? When recently proving that Inn$(G)$ $\le$ Aut$(G)$ we needed to choose two elements $C_x ,C_y \in$ Aut$(G)$ and show the composition, that is $C_y \circ C_x = C_{xy}$, is in Inn$(G)$ too. To do this we have to appeal to something slightly new, and slightly uncomfortable for me: we start by examining $C_y \circ C_x \;(g)$ . I am immediately struck that we are looking to see how this composition acts on an arbitrary $g \in G$ . Eventually we discover $C_y \circ C_x \;(g) \; = C_{xy}(g)$ .
Questions:
We took two elements in Inn$(G)$ that make no sense without a set $G$ to "act" on. For any group $K$, does $K$ always need a set to "act" on? If so, for instance, what set am I shuffling around when I play with $\mathbb{Z_6}$? Is this what is meant by group actions? 
 A: Groups can be defined independently of any group action by the standard group axioms.  
That being said, one way of defining a group action is as a homomorphism $\alpha : G \rightarrow \text{Sym}(X)$ where $X$ is some set.  The group $G$ then acts on $X$ by $\alpha(g)(x)$ for each $g \in G$ and $x \in X$.  
Every group $G$ automatically acts on itself ($X = G$) by the inner automorphisms (as you noted).  Formally, this corresponds to the homomorphism $\alpha : G \rightarrow \text{Sym}(G)$ where $\alpha(g) = C_g$.  Another group action is simply the group operation.  Formally, this is the homomorphism $\beta : G \rightarrow \text{Sym}(G)$ where $\beta(g)(x) = g x$ for each $g, x \in G$.
A: Regarding just your last question:

Is this what is meant by group actions?

Yes and no. The definition of a group action is rather permissive, so the concept is more general than what you probably have in mind. For example, we have:
Super deep theorem. Given any group $G$ and any set $X$, there exists an action of $G$ on $X$.
Proof: Take the trivial action, $gx=x$ for all $g\in G$ and $x\in X$.
But the trivial action is, well, trivial. I bet it's not what you're really interested in. Cayley's theorem is a nicer result because it tells you that every group has a regular action on a set.
