From a categorical point of view, sheaves are considered on a Grothendieck site. If you want to start with the general setting, I recommend the following comprehensive reference in which mainly the properties of the category of sheaves are investigated. Moreover, all of the needed categorical materials are discussed.
- S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag, 1992.
As for the application, you have to choose the desired site, e.g., that of open sets and that of Euclidean spaces. Of course, one of the main applications of sheaf theory in algebraic topology and differential geometry concerns cohomology theories. See, e.g.,
G.E. Bredon, Sheaf Theory, Graduate Texts in Mathematics, Vol. 170, Springer-Verlag, 1997.
Frank W. Warner, Foundations of differentiable manifolds and Lie groups. Vol. 94. Springer Science & Business Media, 1983. (Chapter 5)
L.I. Nicolaescu, Lectures on the Geometry of Manifolds, World Scientific, 2008. (Chapter 7)