Be careful. You need to show that G is closed, not S6 (obviously S6 is closed since it is a group). The operation on S6 is function composition. To show that G is a group is trivial. Since all you have is powers of sigma (I will now denote by s) (s^n)(s^m) = (s^r) where r = m+n (mod 6). Thus it is clear that every element has an inverse since (s)(s^5)=s^6=e, (s^2)(s^4)=e, and (s^3)(s^3)=e.
Also G is abelian since it doesnt matter which permutation you take first, they are just powers of sigma. The last part is easy as well, clearly s^3 is the only element of order 2. The elements of order 3 are s^2 and s^4.