If G is a torsion abeliean group
I'll prod you in the right direction with 1., and then we'll work through 2. TOGETHER, ok? You know that $\displaystyle M\cong R^m\oplus M_{p_1}\times\cdots M_{p_n}$ and so by the way Hom reacts with products
$\displaystyle \displaystyle \text{Hom}_R(M,M)\cong \prod_p \text{Hom}_R(M_p,M_p)\oplus\prod_{p\ne q}\text{Hom}_R(M_p,M_q)\oplus \prod_p \text{Hom}_R(M_p,R^m)\oplus\prod_p\text{Hom}_R(R^m ,M_p)$
So, now, tell me, why do the last three direct summands (groups of direct summands) dissapear?
Well, that is true in some sense, by why does that help us? What we really want to show is that each of the three Homs $\displaystyle \text{Hom}_R(M_p,M_q), p\ne q$, $\displaystyle \text{Hom}_R(M_p,R^m)$, and $\displaystyle \text{Hom}_R(R^m,M_p)$ can be reduced. How can we reduce each of these?