Determine $\text{Vect}_k(S^n)$ Let $\text{Vect}_k(X)$ denote the isomorphism classes of rank $k$ real vector bundles over smooth manifold $X$. 
Is there a rule for determining $\text{Vect}_k(X)$ over reasonably nice manifolds?
Specifically, how to calculate $\text{Vect}_k(S^n)$, for $n$-spheres?
 A: As a general question, this is not an easy question to answer at all. It is studied in the field of "topological K-theory". Much of the work in this field is to answer an easier question of how many "stable equivalence classes" of rank $k$ vector bundles over $M$ there are, where two rank $k$ vector bundles $E_1 \mapsto M$, $E_2 \mapsto M$ are stably equivalent if there exists some trivial vector bundle $\epsilon \mapsto M$ of some rank such that $E_1 \oplus \epsilon$ and $E_2 \oplus \epsilon$ are isomorphic. And this easier question is still very hard to answer.
Still, though, there is a topological rule (which underlies the $K$-theory point of view). There is a "classifying space" for rank $k$ real vector bundles over $m$-dimensional manifolds $X$, which means that the isomorphism classes of such bundles are in 1--1 correspondence with homotopy classes of continuous maps from $X$ to the classifying space. This classifying space is the Grassmanian of $k$-planes in $\mathbb{R}^n$ for some sufficiently high value of $n$; I forget how high $n$ has to be, but it has the form $n \ge C(k,m)$ for some constant depending only on $k,m$. 
Your question about spheres then comes down to classifying homotopy classes of maps of spheres to the Grassmanian. This converts your question about spheres into a question of homotopy theory, namely computation of homotopy groups of Grassmanians, which makes algebraic topologists happy.
A good resource for all of this is the book "Characteristic Classes" by Milnor and Stasheff, and some of the more advanced books on manifolds such as "Differential Topology" by Hirsch.
