The modern theory of vector bundles over a curve starts with Grothendieck's article Sur la classification des fibrés holomorphes sur la sphère de
Riemann. American Journal of Mathematics, 79, 121–138, 1957.
(Actually Geyer and others afterwards realized that the gist of the theorem, in another formulation of course, goes back to results of Dedekind-Weber back in 1882! See Scharlau's article below)
Atiyah then classified, also in 1957, vector bundles on an elliptic curve in this splendid article.
There was then in the next decades an intense activity in getting results on moduli spaces for curves of genus $\geq 2$.
Leaders in the field were among others Narasimhan and Seshadri and here too one can find older predecessors, notably André Weil with his 1938 article article Généralisation des fonctions abéliennes.
Some pleasant didactical references :
For the classification of vector bundles on complete curves in the spirit of Geometric Invariant Theory there is Chapter 5 of Newstead's book
Here is a great survey by Newstead, one of the creators of the theory.
Scharlau's history of Grothendieck's classification. And a proof of that classification can be found on page 23 of Montserrat Teixidor's survey (which by the way is one of the best texts I can recommend as an answer to your question).
An elementary survey by Cautis, emphasising the comparison of holomorphic, and topological vector bundles on Riemann surfaces.
And finally, another survey by one of the historical masters of the field.