A collection of Isomorphic Groups 
So the answer is

Questions
1) How exactly is $<\pi>$ isomorphic to the other integer groups? I mean $\pi$ itself isn't even an integer.
2)What is exactly is the key saying for the single element sets? Are they trying to say they are isomorphic to themselves?
3) How exactly is $\{\mathbb{Z_6}, G \}$ and $\{ \mathbb{Z_2}, S_2\}$ isomorphic?

Jacob Schlather has already answered your first question, spelling out, you can consider the homomorphism $\varphi \colon \mathbb Z \to \langle \pi \rangle < \mathbb R^*$ given by $\varphi(n) = \pi^n$ for each $n \in \mathbb Z$.
About your second question, the exercise asks you to divide the given family of groups in isomorphism's class (i.e. it require to describe the equivalence classes for isomorphism relation inside the given family of groups).
Finally, addressing your third question, the group $G$ is generated by an element (it's a cyclic group) which can be written as product cyclic factors $(1\ 3\ 4)(2\ 5)$ so it's a cyclic-group generated by an element of order $6$ (luckily there aren't many groups of this type :) ). 
Similarly you have that the group $S_2$ is a group of order $2$ and also in this case there aren't many group of this type.
