Gaining insight into the Inverse Image Sheaf Let $f: X \rightarrow Y$ be a continuous map of topological spaces and let $G$ be a sheaf of sets on $Y$. I am trying to understand the definition of the inverse image sheaf $f^{-1}G$ on $X$. This is the sheaf associated to the preshief $U \rightarrow \varinjlim_{V \supseteq f(U)} G(V)$ where $V$ is open in $Y$ and $U$ open in $X$. In particular i am trying to understand the quantity $\varinjlim_{V \supseteq f(U)} G(V)$. As i understand, this is a colimit in the category of sets. I read the category-theoretic definition of the colimit using co-cones, but i am having a hard time making a connection/interpretation. Also, is this a filtered colimit or just a colimit? 

1) First of all you have the crucial property $(f^{-1}G)_x=G_{f(x)}$ for all $x\in X$.
In particular, applying this to the inclusion $i:\lbrace x\rbrace \hookrightarrow X$, of a point, you get the interesting equality $(i^{-1}G)_x=G_x$ which shows that taking an inverse image is a generalization of taking the stalk of a sheaf at a point.  
2) This is a situation where the pre-Grothendieck interpretation of a sheaf as an étalé space is quite illuminating:
If $Et(G)\to Y$ is the étalé space corresponding to $G$, take the fiber product $Z\stackrel {def}{=}X\times _Y  Et(G)\to X$ in the category of topological spaces.
This is an étalé space over $X$ and the corresponding sheaf of sections is the sheaf on $X$ you are looking for: $Sh_Z=f^{-1}G$.  
3) Beware that in the category of schemes (or in any other geometric category like that of analytic spaces) the important pull-back for sheaves of $\mathcal O_Y$-modules over $Y$ is not the topological one we are discussing but the algebraic one, namely $$  f^{*}G      =f^{-1}G \otimes _{f^{-1}(\mathcal O_Y)}  \mathcal O_X     $$
The simplest contrasting example is that of a scheme $X$ over a field $k$ and the inclusion of a rational point $i:\lbrace x\rbrace \hookrightarrow X$. There is a huge difference between both pull-backs of $\mathcal O_X$:
$$  (i^{-1}\mathcal O_X)_x =  \mathcal O_{X,x}\neq     (i^{*}\mathcal O_X)_x  =k                $$
