Computing $\mathrm{Vect}_k(M)$?

Question

$\mathrm{Vect}_k(M)$ is the isomorphism classes of real $k$ rank vector bundles over $M$. In Bott-Tu book they give only an example: For contractible manifolds it is $\mathrm{Vect}_k(M)$ is a point. I don't think I could understand well only by this example. Could you give me other nontrivial examples? In Exercise 6.10 of this book. It is asked: Compute $\mathrm{Vect}_k(S^1)$.