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](http://doi.org/10.2307/2372388). 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](https://math.berkeley.edu/~nadler/atiyah.classification.pdf) 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](https://gallica.bnf.fr/ark:/12148/bpt6k6459126x/f59.image)*.

**Bibliography**   
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](http://books.google.fr/books/about/Lectures_on_introduction_to_moduli_probl.html?id=1yAiRAAACAAJ&redir_esc=y)     

[Here](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=75&cad=rja&uact=8&ved=2ahUKEwi-jvTu8MLeAhXETrwKHXx_AKc4PBAWMA56BAgGEAI&url=https%3A%2F%2Fwww.mimuw.edu.pl%2F~jarekw%2Fpostscript%2FLukecin-Newstead.ps&usg=AOvVaw2RdJIOkENH0RQI7CAWz8zW) is a great survey by Newstead, one of the creators of the theory.    
 
[Scharlau's history](https://ivv5hpp.uni-muenster.de/u/scharla/scharlau/grothendieck/Grothendieck.pdf) of Grothendieck's classification. And a proof  of that classification  can be found on page 23  of  [Montserrat Teixidor's](https://pdfs.semanticscholar.org/dd79/abcec738d08efe948e4a9aa5ad74dacf16e2.pdf) survey (which by the way is one of the best texts I can recommend as an answer to your question).  
[An elementary survey](http://www.math.ubc.ca/~cautis/classes/notes-bundles.pdf) by Cautis, emphasising the comparison of holomorphic, and topological vector bundles on Riemann surfaces.  

And finally, [another survey](https://www.cimat.mx/Eventos/c_vectorbundles/ramanan_notes.pdf) by one of the historical masters of the field.