I would like to find some notes (or book), that explains on a very basic level what is a line bundle on a complex projective manifold. Maybe even, what is a line bundle on $\mathbb CP^n$. It seems to me that in a "usual" algebrogeometric approach line bundles come quite late, only after one defines what is a sheaf, ect. I wonder if this can be explained in "quicker" way?

If you want complex vector bundles (or even complex fiber bundles) presented at a "very basic level" (without sheaf theory), try Chapter IV of Fritzsche, Klaus; Grauert, Hans: From holomorphic functions to complex manifolds. Graduate Texts in Mathematics, 213. SpringerVerlag, New York, 2002. xvi+392 pp. ISBN: 0387953957 


For a slightly different approach, I recommend Algebraic Curves and Riemann Surfaces by Rick Miranda. This book starts out by developing everything in terms of divisors, then turns to line bundles and sheaves. I recommend it precisely because it works at developing lots of motivation for the notion of a sheaf (as well as that of sheaf cohomology). 


I think there are several good references. For instance: R. O. Wells: Differential Analysis on complex manifolds (Springer GTM 65), Chapter III. J. D. Moore: Lectures in SeibergWitten invariants (Springer LNM 1629), Chapter 1. D. Huybrechts: Complex Geometry  An introduction (Springer Universitext), Chapter 2. 

