How to prove a vector bundle is non-trivial using transition functions

My knowledge on this topic is a bit sketchy. I realize that there is a whole branch of math out there devoted to checking whether a certain vector bundle is trivial or not, and I partly know some stuff about characteristic classes that seems to do just that (the cases of the Euler classes and Chern classes are well-known to me).

But what if we define a vector bundle in terms of its transition functions? Is there a natural framework to check whether it's non-trivial? I know a bit of Cech cohomology, and it seems that the cocycle condition satisfied by vector bundle makes this piece of math quite relevant. I know of it being used for holomorphic line bundles, but I don't know of a more general approach where one calculates some related Cech cohomology for a general vector bundle given in terms of its transition functions.

Can anyone point me into the right direction? It's not even that I need to calculate something, I'm more generally interested in how one would approach this problem.