How to prove this theorem?
A finitely presented module is projective iff it is locally free (the localization at every prime ideal is free over the localized ring).
This is given right after the statement that a finitely generated module over a local ring is free. How to use it? If $M$ is a finitely presented projective module, then its localization $M_P$ is a module over a local ring. Is the localization again projective? If so, the statement I mentioned can be applied.
For the other direction, the hint says to use that if $S$ is flat over $R$ and $M$ is finitely presented, then there is an isomorphism $$S\otimes_R Hom_R(M,N)\to Hom_S(S\otimes_R M, S\otimes_R N)$$ More specifically, the hint is to show that for any surjection $\phi: F\to M$, the map $Hom_R(M,F)\to Hom_R(M,M)$ is also a surjection. But not only don't I see how to deduce that from the above, but also I'm not sure why this is enough to show.
$M_P$ is projective for every prime $P$. This follows from the fact that $M_P \simeq R_P\otimes_R M$, and that $R_P\otimes_R -$ is left adjoint to the forgetful functor from $R_P$-modules, which is exact, therefore it preserves projectives (you can give a proof of this by hand of course if you don't know enough about functors).
For the second question, note first of all that the second hint is enough to conclude : indees it implies (by taking an antecedent of $id_M$) that any short exact sequence ending with $M$ splits, which is another definition of projective.
Now note that for $R$-modules $A,B$, $A\to B$ is surjective if and only if $A_P\to B_P$ is surjective for any prime $P$ (that's a classical lemma). Therefore if you start with a surjection $\phi : F\to M$, you may want to study $\hom(M,F)_P\to \hom (M,M)_P$. But now, $A_P= R_P\otimes_R A$ for any $A$ so what does the first hint tell you ?