I personally won't recommend Bredon's book, rather Iversen's "Cohomology of sheaves" (especially if you are interested in the topological aspects/applications of sheaf theory).
There is also Dimca's "Sheaves in topology". However I should say that the epigraph to this (very good) book is "Do not shoot the pianist", and maybe not without a reason.
If you are more into algebraic geometry, then you should read chapter 2 of Hartshorne.
A classical reference is Godement's "Topologie algebrique alge\'ebrique et th\'eorie des faisceaux".
Prerequisites for all of these are some algebra (the definitions of a ring and a module, basically, but if you've never seen complexes before, you may find the presentation a bit dense in the beginning)beginning; you'll also need some commutative algebra if you are reading Hartshorne), some basic general topology , (and also some theory of smooth manifolds(, e.g. partitions of unity, in the case of Iversen's book) and some category theory. You could just start reading Hartshorne or Iversen (depending on what the goal is) and then look up categorical notions that are unfamiliar in MacLane's "Categories for the working mathematician" or on Wikipedia.