I'm trying to show that the functor $\Gamma(X,-)$ from the category of quasi-coherent sheaves maps a quasi-coherent sheaf to an R-Module, and also that for coherent sheaves the same functor takes the sheaf to a finitely generated R-Module (where the sheaf is defined over a Noetherian affine scheme.)
However I'm struggling to see why this is true.
I have that for the structure sheaf we can set $X=Spec(R)=U_1=Spec(R)$ \ (V(1)). Then $O_X(X)=R_1=R.$ Where $R_1$ is the localisation at 1.
Hence $\Gamma(X, O_X)=R$ for $X=Spec(R)$
But am stuck as to where to go from here to do the case for quasi-coherent/coherent sheaves over a Noetherian affine scheme. Can anybody help?
There are several equivalent definitions of coherent sheaves and as the comment pointed out, some of those trivially addresses aspects of your question.
If you've defined (quasi)-coherent sheaves on affine schemes to be the $\mathcal{O}_X$ module $\tilde{M}$ satisfying $\tilde{M} (D(f)) = M_f$, as in Liu's book, then this is immediately verified.
If you wish to proceed as in Hartshorne, where $\tilde{M}$ is locally defined to be the set of functions from your open set to all the stalks of points in that open set which is locally a fraction, then you need to do more work. Hartshorne covers this in Proposition II.5.4 which immediately gives this result and more, which is in Corollary II.5.5. The arguments, including that of Lemma II.5.3 is an algebraic geometry version of the partitions of unity argument commonly seen in differential topology. As such, you may wish to read at least one of the proofs.