How to properly use GAGA correspondence

First let me emphasize what Serre did : in 1956 he wrote an article Géométrie algébrique et géométrie analytique, which everybody calls GAGA (I heard him explain that he deliberately chose the title so that this would happen!).
This article proves many results, the gist of which is that calculations for coherent sheaves on a projective complex variety $X$ give the same (or isomorphic) results as those done on the underlying holomorphic variety $X^h$.

However he did not consider non coherent sheaves and for good reason: if for example you consider the constant sheaf $\mathbb C_X$ on $X$, it is flabby ("flasque") so that all its cohomology groups vanish: $H^i(X,\mathbb C_X)=0$, whereas of course the cohomology vector spaces $H^i(X^h,\mathbb C_{X^h})$ are not zero in general:
For example if $X$ is a smooth projective curve of genus $g$, then $X^h$ is a Riemann surface and $dim _\mathbb C H^1(X^h,\mathbb C_{X^h})=2g $.

Fortunately thanks to the rich techniques known for holomorphic manifolds many apparently transcendental invariants of a projective smooth manifold can be computed using only coherent sheaves.
A splendid example is Dolbeault's 1953 result for Betti numbers $$b_r(X^h)=\sum_{p+q=r} dim_\mathbb C H^q (X^h,\Omega_{X^h}^p)$$ Since the right hand size is cohomology of coherent sheaves,it can be calculated algebraically thanks to GAGA: $$ dim_\mathbb C H^q (X^h,\Omega_{X^h}^p)=dim_\mathbb C H^q (X,\Omega_{X}^p) $$ In particular, for a smooth projective curve $X$, the genus of the associated Riemann surface $X^h$ is given algebraically by $$g(X^h)=\frac {1}{2}(b_1(X^h))=\frac {1}{2}(dim_\mathbb C H^1 (X,\mathcal O_X)+dim_\mathbb C H^0 (X,\Omega_{X}^1))$$ where the two summands on the right are equal by Serre duality (him again!).

Warning This answer is meant to address your request "it would be immensely appreciated if you could elaborate a little on how to use GAGA in general ", not to show the shortest or most elementary road to technically answer your questions on Riemann surfaces.