I intend to try working through Bredon's seminal sheaf theory text prior to graduating (I am currently a second year undergraduate), but it is at a level which is far beyond my own (friends of mine who study algebraic topology have gone until their second and third years as graduate students before touching it). However, I am interested in algebraic geometry (though the material treated in Bredon's text is certainly of relative interest to me) and find Bredon's "Topology and Geometry" to be a superb treatment of the algebro-topological tools which may have some utility in my future studies (Bredon takes a more geometric approach). Is there another text which might be a better 'crash course' on algebraic topology for someone at my level (a bit of algebra, analysis, and point-set topology, with a good deal of category theory), or am I on the right track with Bredon's text? Thanks!
Firstly, as you say you are interested in algebraic geometry, Bredon's book may be a slightly unfortunate choice. It very much emphasizes the point of view of the "espace étalé"; it's not much harm to translate things back into the "site" perspective, which is the only one that generalizes to algebraic geometry.
No offence to this great book, but it is extreeeeeeeeeeeemely technical and certainly written for people who want to know all the possible subtleties of sheaves on topological spaces (e.g. it is full of beautiful examples that show how badly things can go wrong ;-); but that's certainly not the kind of stuff a beginner in algebraic topology should learn; and needless to say, it's about subtleties in topology that the Zariski topology certainly does not have.
My advice would be the excellent introduction by Demailly in his online book http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf - the chapter about sheaf cohomology is essentially self-contained and avoids the use of higher homological algebra (unlike the books of Shapira / Kashiwara etc., Schneiders).
Have fun! Sheaves rock!
I think you are on the right track in the sense that it's a good idea to read Bredon's Topology and geometry while you're an undergraduate. It starts from scratch more or less and contains the necessary prerequisites for many more advanced books in different areas: algebraic topology (Spanier etc), differential geometry (Besse), Lie groups, you name it.
On the other hand, I don't think that it is formally a prerequisite to reading Sheaf theory. To do this you only need some point set topology and some basic homological algebra (complexes and cohomology but no abelian categories).
On yet another hand (how many hands can one have?), chances are you'll never need to read Bredon's Sheaf theory -- Godement or Iversen are (in my opinion) far more suitable as a first introduction to sheaves than Bredon. There are also two books by Kashivara and Schapira. One is "Categories and sheaves" and the other is "Sheaves on manifolds". Both develop the theory from scratch (especially the first one) but they are a bit dry for a first reading. In fact, the only thing that you can find in Bredon's Sheaf theory but not in any other introductory text I know of is the Smith spectral sequence.
Since this is already way too long to for a comment, let me mention some other sources similar to Geometry and topology that you may find useful. Bott and Tu's Differential forms in algebraic topology can't be recommended high enough. This book starts at an elementary level but contains some of the key ideas in topology, which come up over and over again in many contexts. There are no sheaves there, but if you read it carefully (i.e. try to reconstruct all the proofs in detail), the definition of a sheaf will come as no surprise to you afterwards. Or you may even discover it yourself.
Milnor's Morse theory is arguably one the best introductory mathematical texts ever written and it also falls into the category "what every geometer or topologist should know".
Lee's Introduction to smooth manifolds is quite long but it presents the material in excruciating detail.
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Although it's one of my favourite books, Kashiwara and Schapira 'Categories and Sheaves' is incredibly dry and abstract (and somewhat nonstandard in notation in many points).
When I first learned the material I found extremely valuable the seventh chapter of Taylor's Several Complex Variables with Connections with to Algebraic Geometry and Lie Groups. It's an all round great book and it made me understand things much more clearly.
You're supposed to read it after chapter six (which might be a little bit of an overkill) but if you're not too much into abstract nonsense you can substitute 'sheaves' or 'modules' whenever he speaks of objects in an abelian category.