Dear Mohammad, there is a rather elementary book Introduction to Moduli Problems and Orbit spaces by P.E. Newstead which will explain to you why stability is important, give you lots of examples (Chapter 4 is devoted to them) and which ends with a whole chapter (Chapter 5) called Vector bundles over a curve. It was written by an extremely competent expert and deliberately maintained at a quite elementary level. The author explains in the preface that his notes are an introduction to Mumford's Geometric Invariant Theory in the language of classical algebraic geometry, deliberately eschewing schemes.
On the subject of holomorphic bundles over $\mathbb P^n(\mathbb C) $ you may check Okonek, Schneider and Spindler's monograph Vector Bundles on Complex Projective Spaces, written in the language of holomorphic manofolds (the results are the same as in algebraic geometry thanks to Serre's GAGA principle).
I'd also like to mention Atiyah's classic Vector bundles over an elliptic curve
published in 1957, which I still find quite instructive despite its venerable age.
And finally I should also mention the articles on moduli of vector bundles over curves written by the brilliant Indian school around the Tata Institute: M.S.Narasimhan, Seshadri, Ramanan, Nori, ...