I have two related sorts of example to suggest, probably exhibiting my categorical bias vs. the analytic/geometric-topology weight of the preceding examples.
Let $P\in K[x]$ be an irreducible (separable) polynomial of degree $n$ over a field $K$, and consider the extension field $L = F[x]/(P)$. It is fruitful (i.e., it's the whole categorical view of Galois theory) to view $L$ as one of serveral isomorphic extensions, with possibly non-trivial automorphism group relative to $F$. In one sense, everything you might wish to know about $L$ and $K$ is contained in your ability to understand $K$ and $P$, but it's evidently fruitful to look at commutative diagrams $L//F\to L'//F$
So you want to understand some algebraic gadget? A vector space/lie algebra/group? Then study bundles of such things!
For example, it can be a bit stifling to think of a Lie algebra of a Lie group as the tangent space at the identity. Sometimes it's better to think of it as either space of half-invariant vector fields; and then the Lie bracket is given by the Lie bracket! (ha-hah!) But there is plenty of room to study other things, exploting the bundle of isomorphic Lie algebras.
Other times it's helpful to study the universal bundle $EG\to BG$ of some discrete group $G$. These realise the group both as a particular fiber and as the fundamental groups group of $BG$, BG$ at your favourite basepoint, and so you learn things about the group (e.g. related homology functors) by studying the whole collection of groups $\pi_1(BG,b)$ and torsors $EG_b$ for $b\in BG$.